it is normal for any 'standard' (not limited slip) differential to turn/power which ever wheel turns easiest, so it is not abnormal to only have 1 wheel turn while in the air. what happens if you jamb a board under the 1 wheel that turns easiest? the other wheel should start turning. if it does not, then you have something wrong with that wheel, my instant wag would be that one of the brake shoes or related hardware in the wheel brakes has become defective and will need to be replaced. pull off the drum and inspect the wheel brakes and park brake cables for proper operation.
Can you describe for us a little more what you mean by "not engage the right way"? you may have a broken axle shaft.
in any case, I would want to visually inspect the brakes on any older lift I put into service and did not know any history about, and pulling both brake drums off and doing an inspection to be sure the truck was going to be safe to use will give you the information about the brakes and axles you will need.