I have a ezgo 36 it's got power but runs slow, batteries are full

loose power connection or bad ground somewhere. get a little metal wire brush (or several of them), disconnect battery, and disconnect ground and power wires (keeping track of them), and start brushing them, then re-attach and test

Have the battery checked to confirm a good charge and strong cell...then if all good, you may want to replace this:

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It may need a repair

How old are the batteries? Carefully check the fluid level of the batteries. Wear gloves and full face protection if possible as batteries are filled with corrosive acid and there is a chance of explosion. If the cart sat with dead batteries for a extended period of time without being charged the cells in the battery will harden up due to the low charge state making it hard for the battery acid to chemically react with the lead cells when you are trying to charge them. Try replacing the batteries and making sure all of your contacts are clean and are snugly secure.

Need to know what kind of cart you have. Ezgo, Club Car?

36 volt, 48 volt ? for ezgo look at manuf. code, for club car serial number has the year. Then maybe we can figure out something.

Maybe you have an Ezgo cart? Black is negative, white is positive, and the small red wire is connected to the white inside the charger receptacle.

have the batteries individually load tested as I suspect that there are dead cells in one or more.

First of all, lift rear axle off the ground and place blocks or jack stands underneath. Check to see if you have 36 volts On your batteries. Now you want to run a jumper across the big cables on the solenoid. Press the accelerator and the motor should engage. Clicking is a pretty sure sign that the solenoid is bad. Please let me know if this is useful. I can further elaborate more troubleshooting if this doesn't solve your problem.

He may not have known to put it in tow mode and doesn't want to get blamed for damage to transmission towing it in wrong gear.

Your batteries have died. The reason it will barely move in reverse is that current is being supplied to the motor is a direction that it spends very little time using. There is a slight change in the armature over time, and reverse tends to work better over time. Check the voltage when under load. (run jumper wires from the + and - of the battery bank. If it is a 36 volt cart and the voltage drops to below 36 volts, you do not have enoug current to run the cart. Plug in the charger and watch the ampmeter. If it goes up to about 20 amps, then it is charging. It should take a good charge after 10 hours. If the cart charges and only runs a short distance, the batteries need replacing.