"Chess is war over the board", Bobby Fischer once said.
But chess is also work. If you enjoy the thrill of playing chess, including the risk of blundering, but don't feel up to the task of the work involved to get better at it, don't worry: welcome to a fairly large club.
Engines are labor-saving devices, first and foremost. Unfortunately, the more labor you save, the more out-of-shape you get. Pretty soon you're like the robotically-attended cruise guests in the movie Wall-E, unable to get up out of their levitating deckchairs.
So, if you want to avoid that, you have to go on the hunt for a way to take advantage of what an engine can do for you without giving it all of the heavy lifting to do for you. That's what you're "check-the-engine-after-each-move" technique seems to be doing, except that you're not following up with analyzing your games. (And one other thing, too; more on that below...)
I would bet you can't find a chess coach anywhere who will tell you that you can get better at chess without putting in some time analyzing your games. They'll all tell you that you'll improve more and faster by getting knocked down for the count in about half of your games, and then studying your losses, because that's the most effective way to learn.
After all, that's what the rating system is supposed to do; it puts people of roughly equal ability in a fair match, where either one could win. That means (excluding draws) either player could lose. Therefore, half of the time, you should lose.
a) Unless your partner is doing what you're doing and you both know what's going on, you're tipping the scales and not disclosing it to your partner. Yes, that's unethical.
b) You are depriving yourself of being able to play better, by avoiding the work required to get there.
What kinds of work you need to do to improve:
- Diagnose your strengths and weaknesses. To figure these out, you need to analyze your games. Do it alone, with a friend, or with a coach, but you'll find that you make the most progress if you do it alone first. There's lots of good guidance on how to analyze your games.
Wait to use an engine until after you've done the first pass on your own. Try to find the points in the game where one side improved, and the other side got worse. Try to figure out which move caused the problem, and why. Take notes. Only then, use the engine to discover what you missed in the analysis, and to confirm what you found.
Catalog the kinds of mistakes you make. What are your weaknesses? Which ones cost you the most games? Are they: Blunders? Strategic weaknesses? Poor development? Poor king safety? Unsound sacrifices? Lack of a sense of danger? Allowing counterplay? Poor defense? Lousy attacking? Floundering in the endgame? Not knowing what to do in the middlegame once the book moves are over in the opening? Messing up the opening itself?
Figure out where to start. It's usually either the most common problem, or the one that kills the game instantly when it appears. Then, devise a plan to correct the problem. You will probably need advice from better players or a coach to help with this. Here are some common problems and some possible solutions:
4a. If you miss tactical opportunities (a common weakness among players below about 1800), practice tactics problem sets, and track your progress.
4b. If you allow tactical opportunities, pick up a fresh tactics problem set, and for each one, turn the board around, remember that it's now the other side to move, and figure out what your opponent could do to you. Then try to figure out how you could prevent it, if it were your move. See the New Art of Defence in Chess, by Andrew Soltis, or Colin Crouch's How to Defend in Chess.
4c. If you have a tendency to blunder (miscalculate, overlook a simple capture or immediate tactic, etc), get a book on avoiding blunders; both Angus Dunnington and Lev Alburt have written one, and they're both good resources.
4d. Do you find yourself missing good candidate moves, or misjudging which move was the best, or missing a tactic that came up after the line you analyzed was played out? These are all analysis mistakes, and a disciplined method of analyzing at the board will help. Try Per Ostman's Your Best Move, or Soltis' How to Choose A Chess Move, to solidify your thinking technique.
Then, once you've fixed that problem, Rinse, and Repeat with Step 3.
4e. If you frequently get into time trouble, work on i) budgeting your time per move, ii) knowing when to have a long think and when not to, and iii) finding the best available move when short of time. Blitz games are not a remedy for this; the right approach is to develop a sense of what kinds of situations allow quick thinking, and which ones don't, and learning how to steer the game in the right direction. I can recommend "What It Takes to Become a Chess Master," by GM Andrew Soltis; it has several sections on finding moves quickly and when to apply the techniques.
Then, play slower games, but work on ending the game with a budgeted amount of time to spare. If you meet or beat the budget, you're improving. If not, you won't lose on time, but you'll know you have to improve. Once you can reliably meet your budget, play a slightly faster time control. Do this until you can play a 5 minute game with 10 second delay respectably well. You'll lose quite a few, but you'll be thinking faster and more efficiently. Note that this approach will work much better if you do 4d) first, to make sure you're using your thinking time as efficiently as you can.
During your game, keep track of the time you spend on moves. When you analyze the game, in addition to finding and cataloging the mistakes, try to figure out where you spent what turned out to be too much time, and why. Did you rehash the calculation of lines multiple times? Did you get anxious when faced with only poor choices, and just procrastinate? Did you get distracted?
You can't justify doing any of this work unless you are confident that it will help you improve, and that doing this particular work will be the most help to you now. Many players simply decide they need to work on openings because the lost a quick game. Or that they need to work on endgame theory because they missed a fork in a knight-and-pawns endgame. They could be right, or they could be wrong. If they're wrong, they'll end up spinning their wheels and wasting effort, and will continue to struggle to improve. This happens all the time.
To justify this work, you need that analysis of your games. Set aside time to do it, ideally soon after you play the game. If you wait, you'll forget key information that could provide insight into why you made a particular mistake. If you play multiple games before you get around to it, you might find the amount of work daunting and never get around to it. It's a simple choice: Procrastinate, or Improve. You can't have both.
Another potential issue you mention is that you get anxious not knowing whether you've made a mistake. I'm not a psychologist, and I certainly don't want to read too much in between the lines here, but risk is inherent in everything we do. Sure, it's nice to minimize it, but in the end, we need to get accustomed to accepting risk, and dealing with failure.
On top of the other things it is, chess is also a game. Unless you're trying to make a living at it, it might help to try to develop a greater tolerance for blundering, possibly not finding out that you've blundered until your opponent takes advantage of it, and losing. You should want to win, or you won't be motivated, but you will play better if you're not afraid to lose.
Postscript on Fear of Failure: In his article "An Improvement Plan" for ChessCafe.com back in 2006 which I recently rediscovered, Dan Heisman said: "...results will be strongly diminished if you can’t abandon your fears about losing and your rating. Any good improvement program will include enough practice that you will be faced with plenty of losses and the need to endure times when your rating goes down. That’s life; it happens – no one goes straight up. So if multiple setbacks cause discouragement and not determination, it’s going to be one long path…"
You will learn much more from your mistakes than from perfect play. Make the mistakes, and then learn from them. Play, on your own, and study your games afterwards.
You will improve fastest if you can persuade yourself to do those things. Good luck!