It really is a fun summer bag ... do don't beat yourself up with disappointment!
Thinking the same as some of the others ... load it up and see how you feel about it. Sometimes I like the bags that have some give to them, as yours does, rather than being too stand-up-straight-and-sturdy!!
For the linings ... sometimes I get it straight and in place, pinning well, then run a few lines of stitching on the bottom. Around the box base or just straight rows across the bottom or even just one line across the length. It's kind of awkward to squash the bag all down to get it into the sewing machine and then stitch, so that often determines what and how much i do. But this can hold your lining in place a bit better.
Noticed that you do not have pockets in your lining or on the outside. Another way to get a bit more stability would be to quilt the outside and lining all as one sandwich, as you would a quilt. Then stitch it together. Yes, this gives raw side seams ... but you can either stitch it to the outside or inside, then finish with a binding. Likewise, this method would require a binding around the top.