I'm looking for chess software (free or otherwise) that will allow me to completely clear the board and set it up as I wish in order to analyze a specific situation. All of the chess puzzle software I've seen seems to have only pre-loaded positions to solve.
You don't mention what platform you are using, but on Android I use the "Analyze This" app - it lets you set up positions from scratch, and also use an engine to analyze. There is a free version of the app, and a paid "Pro" version. In the free version, the only available engine is Stockfish; in the "Pro" version, you can add other UCI engines.
Most general-purpose chess software has this feature, usually under "Edit Board" or similar menu entries.
On Linux, usually there are two components to the software which are developed independently - a board program and a playing engine; the latter acts as an opponent or an analyser. The board program, such as Scid or Xboard, is where you would look for the board setup functions.
On Windows, the popular chess engines usually come with their own board front-ends. Crafty is a decent free option.
It's also possible to use an online browser-based platform such as LiChess.