Replaced all three stems on tub/shower and replaced seat on diverter valve... water still runs from tub spigot and shower at the same time

Hi Tlove and welcome to the AMHD.

If you have an exact replacement stem and seat then it is possible that what you have going on here is that the newer shower heads are limited to 2.5 gallons per minute nowadays and back when these valves were popular shower heads were running at nearer say 6-8 gallons per minute. In other words, with the restriction at the newer shower heads the water is simply not able to get out of the shower head and you get a resultant back pressure that causes the water to push past the new (and old) diverter/diverter seat. Does that make sense?

For fun, you could remove the shower head and test this hypothesis... ;)

If you try the test and it fails then the only other answer is that the replacement parts are NOT an exact match or the tolerances built into the valve itself always allowed for some water loss to the tub spout when water is diverted to the shower head.

That's my thoughts here.

Back to you...

Mark

Assuming that this is a three handle tub and shower valve.
Remove the diverter stem, make sure that it is in the open position (shower) and reinstall it.