In a th350 trans, two clutch packs have to come on for reverse.
The low reverse pack and the direct (or also referd to as third and reverse).
Since you have third gear, your problem is with the low reverse clutch pack located in the very rear of the trans.
You need to cover all of the basic items, such as fluid level, shifter adjustment, line pressure.
Does reverse feel like netural or does the vehicle feel locked up (unable to rev engine freely?)
Your failure does not sound like a simple fix.