It's hard to say what will happen. The floppiness is hard to quilt, so I'd probably correct that and not worry about about anything else until the end.
On my frame, I sometimes take the first pole off the frame (this is the pole that the finished quilt is rolled to). If you did this you might be able to see if the quilt smiles or frowns, or just keeps a stone face.
If this is a bed quilt, I wouldn't worry about it. Just wash the quilt and admire.
If you are quilting a wall hanging or a quilt show entry, you can try blocking it after you are done with the quilting. Blocking corrects a multitude of problems, but it's kind of a pain to do, so I only do it when appearance matters.
I love that you are quilting this yourself - and I bet it looks better than you think!
Janet