Fix 3 handle faucet when screw in handle broke off

[Msfixit]

I have a 3 handle faucet and the screws in 2 of the handles have broken off. I also have a leak in this same faucet so I would like to repair that as well. I had a plumber come out and give me an estimate to replace the existing faucet with a 1handle model (they said that was easier then fixing this one) but I dont have $1300.

I've looked at a number of videos but I haven't seen one that shows what my faucet looks like. I have the handles off but my faucet has a plastic piece before the stem. Not sure what it is called or how to remove it. Part of the screw sticks out but so far I haven't been able to get the screws out.

I've attached a photo.

I would appreciate any help on how to disassemble this so I can repair the leak and be able to reattach the handles.

 

[CountryBumkin]

Can you still remove the handles with the broken screws? If the head of the screw broke off completely, you should be able to remove the handle same as if the screw came out. Then you can replace the cartridge or make whatever repair you were planning on, and then perhaps glue the handle back in place (and next time it needs repair go for the full $1300 replacement job). Sort of a half-ass fix, but it might last for many more years.
Obviously, I'm not a pro.