Before you pull the motor, check the batteries (full pack voltage) under load while driving. Use tape or alligator clamps to affix meter leads to main + and - on the pack, report the voltage before starting, and under load voltage, and note recovery time. Your cart is the best load tester on the market.
The cart should have 4 noticeable "speeds." Solenoid click one is the master, should net no movement. Click two, first gear. Three, second gear. Four, third gear. Fifth and final click, fourth and top gear. If this is the case (you can drive it on flat ground or jack it up and look/listen to test) then there is no reason to mess with the speed slider and microswitches.
Hope this helps! I love these old 5 solenoid carts, dang near bulletproof and easy to troubleshoot most of the time.