Escape brakes grabbing
 

I just replaced the rotor/pads/caliper on the front right side due to a sticking caliper that had caused the pads to wear down to metal and gouge the rotor severely.

The result is that the drivability is much better, but the pads are still sticking a little bit at that corner.

What would cause this? It's as if there is more brake pressure on that corner and it's causing the pads to not disengage completely there.

If all mechanical things have been addressed, you might look into replacing the brake hose for that caliper.

Replaced the brake hose. The brake is no longer sticking on that corner.

But the brake lamp comes on on the dash most times I start the car. Clearing the codes doesn't help, it just comes back.

What code(s) are we getting?

C1145 RF speed sensor
C1155 LF speed sensor
C1165 RR speed sensor
C1233 LF speed sensor signal missing
C1234 RF speed sensor signal missing
U0414 ???

Some ABS systems will send a signal down to all the wheel speed sensors as soon as the key is turned on . This is to verify there are no opens or shorts in the system.
Suggest you recheck all your work, and if you cannot find anything , have a qualified shop check it out more thoroughly..A lot of times something accidentally gets disturbed without our noticing it . The U code is something that needs to be further investigated.

It makes sense to me that it is a signal problem, since I've replaced the LF speed sensor multiple times. On top of that, after clearing the codes they light right back up on the gauge cluster even before the car is put in motion.

Unfortunately as you already know by this time, when a code indicates a particular sensor,it is not always the sensor.
In my experience , 75% of the time it is not the sensor.
All the code does is , provides a direction for the repair person to pursue.

Sometimes it is a part that triggers the sensor and we need to look for those things .
Sometimes it is a wiring or connection problem.
What we do is to get a graph of each sensor and how it is working. If all is well there we go to where the signals go next and verify the signal is getting there
If all is well there, we go to the next part, that provides a response to a correct signal. Could be a module or something else.
Some special equip is necessary to do these checks and most folks don't have the equip or info to verify the next steps..
That is when we need to have the capable shop get involved.