This quad runs fine and shifts into gear fairly well but slips out immediately when there is a trailer hooked on the back or when I am riding and get in it really hard. Ideas where to start looking?

When it slips out it makes the same sound as if you were trying to put it in gear when the motor running too fast.

 

350 was made between 1990-1993. If you've eliminated the outer drive sprocket,chain and excess slack as a problem,more than likely it's the internal silent chain in the gear case jumping teeth when under a load. Plus on these models make sure the red button on the shift lever is locked in when in gear. Can do the same thing and jump out of gear if it isn't.