There are many different designs for limited slip differentials. The limited slip differentials I have owned have all been clutch based and have all lost their effectiveness over the years. I would not pay $1200 to fix a limited slip differential which was probably not very effective in the first place.
If you want to be assured of traction get a selectable locking differential. The ones I am familiar with are ARB air lockers. When engaged the left wheel will always spin at the same rate as the right wheel... the wheels are locked together as if there is no differential at all.
In my experience the best way to improve traction is getting more aggressive tires. Second is a locking differential. Third is four wheel drive. On my trail rig (A 76 Scout with ARB locking differentials front and rear) there are several places I can get in to / out of in 2wd with the rear diff locked that I can not with 4wd and open differentials.
A locking diff is pretty expensive but in the same neighborhood you were quoted for rebuilding your limited slip. If you find yourself getting stuck often it might be worth the cost. You will probably find that you will rarely lock your differential BUT when you need to it can really save your day!
