What’s the best way to patch baseboard trim

You would usually caulk the gap between the baseboard and the wall. This is why I usually don't bother painting the trim before putting it up (unless I'm staining it).

Usually I would prime the wall first, then put up the trim, putty/caulk, paint the trim, then finally paint the walls.

The strips you made and installed are called extension jambs. We use them all the time to fit new windows and doors. I usually sand and prime them before I install them, but if you have already installed the jambs and old trim, here are a couple of ways to finish them off.

1. Put a straight piece of blue painters tape down the wall adjacent to the door trims.
2. Sand the edges carefully so the jambs and trim are flush.
3. Clean off the dust and apply a bead of latex/acrylic painter's caulk.
4. Using a 1" putty knife, smooth the caulk to a nice flat layer against the jamb/trim.
5. Remove the tape immediately. Retouch the caulk gently if removing the tape leaves a ridge.
6. Let the caulk cure for 24 hours. Minor imperfections can be smoothed out with a sharp utility knife and a damp cloth. If there is some shrinkage in the larger gaps, add a little more caulk, smooth off with your putty knife, let cure.
7. Paint.

There are painter's caulk that will "fast dry" in an hour, that you may consider. If you are unhappy with the final leveling of the caulk, you can also use a super thin layer of lightweight spackle (not standard vinyl spackle) applied with a putty knife over the caulk. This product dries in about 20 mins and can be lightly sanded or shaped with a damp cloth.