I recently did this intentionally to a sweater I picked up in a thrift shop. Yes, you do cut up the sweater after you’ve felted it.
I found that anywhere you can see stitch definition, you’re likely to have a bit of fraying/unraveling when you cut, so keep agitating it in the hot water until you can’t see the stitches at all anymore.
Some parts of the sweater (like the main body) felted much faster than others (the sleeves and the ribbing at the neckline and around the bottom). My particular sweater also shrunk more in length than in width. FYI - it was an American Eagle sweater in a pure red and the felt it made was really beautiful.
The only two problems I ran into were, #1, I did use any sort of pattern for the purse and did not measure very well when I cut out the pieces, so the purse was much smaller than I intended and, #2, the sweater I chose was quite thick and I had a hard time getting two layers of it under the presser foot of my sewing machine. I wound up having to handsew parts of it.
Also, I am not the greatest seamstress, so the whole purse turned out a little crooked, too.
It was an interesting experiment, and I learned some good lessons for the next time. Hope my boo-boos help you too!