I'm considering transferring to a school with a worse CS program to avoid 3 semesters of W's (withdrawal grades) and a dismissal on my record come graduation time. (Or would I just have more explaining to do?) Should I stick with my school?
Graduate schools generally ask for all of your undergraduate transcripts, not just transcripts from schools you got a degree from.
Transferring is a decision you'll have to make on your own, but it is probably not a mechanism to clear your record of Ws if that's your intent.
In the meantime, probably best to focus most of your attention on your mental health, followed by graduating itself, regardless of future plans past then.