You could develop a variety of underhand serves -- both topspin and underspin serves. You can use the latter to throw in an occasional drop shot serve that bounces close to the net and does not travel forward much after the bounce. You can also play with sidespin (vertical axis) and spiral spin (aka gyrospin or corkscrew spin) to give your backspin serves more variety. With enough spiralspin (in conjucntion with sidespin & backspin), you can get some serve to bounce wildly off to the side.
With topspin serves you can develop some deep lob serves that bounce high -- a bit like a topspin kick serve. You can vary height and the amount of spin to add variety to these topspin serves. Vary the placement as well. You are not going to get much pace on these serves but the variety could help to keep your opponent a bit off guard.
Not a bad idea to develop overhead serves with your non-dominant arm. It is a great brain development exercise. I have managed to develop some decent serves with my other arm. Start by developing your throwing motion. Perform the throwing motion starting in a trophy position. Throw some balls (with your non-dom arm) upward at a 45 degree angle over the net. After performing a number of these, throw the ball upward at a 75 degree angle. This steeper angle should better simulate the upward swing path of the racket on an actual serve. It should also help you to eventually develop a decent racket head drop (from the trophy position).
Take an an old racket out to the park. Use this as a throwing racket. Throwing a racket feels different than throwing a ball. Start with the racket on your shoulder and/or start with the trophy position. Again, perform some 45 degree throws and some 75 degree throws. Hopefully, you will get a good racket head drop when you start from the trophy position and throw upward at the steeper angle.
Try some hatchet throws -- throw the racket on edge as if throwing an ax or tomahawk. The racket should spin head over tail. After a number of these, start your throws with the racket on edge moving up to the "big L". As the racket head comes up from the "big L", add pronation to re-orient the racket face before you release it. With maximum pronation, try to simulate a flat serve (squaring the racket face to the imaginary ball). Next, employ different swing paths and amount of pronation to simulate spin serves -- slice serves, topspin serves and topspin-slice serves.
One of the challenges of learning to serve with your non-dom arm is developing a decent/reliable toss with the impaired shoulder of your dominant arm. Hopefully, this will not be an insurmountable problem.