Assuming you put the new stem in correctly ( we can talk about that later) then we can mention 2 possibilities:\
1) Water pressure in the house way too high. Easily checked at a washing machine hose connection, with a simple guage available at hardware store.
2) The tub filling up could indicate an not-related drain clog. But if the pressure is too high, it might be overwhelming the drain.
Check your pressure, if OK....take out the diverter stem, inspect, and post photos.