Should I study these openings and try to overcome my aversion?

You should do this. Face your fears don't avoid them. But you have to be very careful here to understand what "study" means in this context. It does not mean memorization, it means understanding. Set yourself a task of finding out what is going on in the first three moves of these openings, then build your understanding from there.

Another approach is to increase your knowledge of strategy and middlegame, then no matter what openings are thrown your way you can be at ease knowing the game is always going to move into an area you are comfortable with.

If you are a beginner I would hazzard a guess that the most efficient approach would be a combination of understanding just the first few moves of these openings then putting most of your effort toward middlegame study.

Should I study these openings and try to overcome my aversion?

You should do this. Face your fears don't avoid them. But you have to be very careful here to understand what "study" means in this context. It does not mean memorization, it means understanding. Set yourself a task of finding out what is going on in the first three moves of these openings, then build your understanding from there.

Another approach is to increase your knowledge of strategy and middlegame, then no matter what openings are thrown your way you can be at ease knowing the game is always going to move into an area you are comfortable with.

If you are a beginner I would hazard a guess that the most efficient approach would be a combination of understanding just the first few moves of these openings then putting most of your effort toward middlegame study.