We want to use one Forms process for the entire training process (moving from phase to phase, completing training, etc). Within each phase of training, the same form is used during each work shift (day) to score the trainee. My back-up is to use a Forms process for each shift, and just submit a blank form, but it seems like we should be able to clear scores and assign a new shift with a blank form. Does this explain the overall process better?

If you ever want a blank form, it usually makes sense to submit a new instance. From what I gather about your process, a trainee spends multiple days on each phase of the training, and each day they have to fill out a scoring form. I'm assuming the scoring form for each phase is different. What isn't clear is where you put the data for the previous day that gets cleared out after submission. 

One solution might be to have a single process with a form made for each phase. The process could be triggered manually each night or automatically with a timer event (not a message start event). The first step would be to run a workflow to determine what phase that trainee is in and what day it is, then assign a user task using the form for that proper phase. The scoring form would be filled out based on the current phase + day, then saved to the repository in that trainee's folder. You'd start a new instance for each day for each trainee, and the same process could handle all forms for each phase and be submitted fresh each day. 

I think conceptually, you should try to work your process around 

start event -> check where the trainee is at (phase + day) -> assign a user task with the proper form for that phase -> save the data somewhere -> update metadata or database with new trainee information (new phase) -> end event

Each day you'd submit a fresh instance based on where the trainee is at.