Really need help to save my 2004 lincoln ls with transmission problem

I just bought a lincoln ls 2004 v6 car with 11,3000 miles. The engine stalled twice when I stopped quickly at a red light signal. I had to push to P and restart the engine again. The engine light is not on and I did not feel it is the misfire problem. Another thing, each time I change from D to R, I can feel my car a little shake (if I do it slowly, the car don't shake). And sometimes when I push the gas a lot when the signal turns green I heard a big noise which sounds like the gear is not matched. I drove to the lincoln dealer to check the engine problem and they took 3 days and they they can't find any problem about the engine but they found the transmission problem. They performed electronic tranmission diagnosis and said EEC test KOEO pass CONT PASS KOER pass. They told me : the transmission fluid level OK and fluid is varnished. They also do a road test and told me: transmission neutralizes AHEN coming to a stop and shifts erratic also makes whining noise on acceleration from torque converter. They suggest me to change the transmission which will cost 3,700 $. I don't want to pay that much money so I came here to ask for help. This is my first car and I like it. I don't know anything about car but I want to save my first car with a reasonable money.

Thank you in advance.