I am also finding text selection in iOS 13 hit/miss. I either cannot activate the text selection or I cannot select without the cursor jumping all over the place. Also, I miss the magnifying glass for text selection. Did Apple remove that?
The cursor will stay in the same place if you move your finger down some so it's not obscuring the selection; on a multi-line input, the first line's selection area extends down a half an inch or so, with all the lower lines displaced by a similar amount.
It's not too bad when you get used to it though I still miss the glass when I'm using my phone or my smaller iPad.