Are you checking the fluid level hot and in neutral? The condition you describe could be low fluid level. The good book also states a broken or misadjusted linkage/cable could cause this. There's a lotta misunderstanding out there about band adjustment, so I'd have that checked. You don't just tighten them down and go. That'll cause them to rapidly fail. Your condition could also be any one of the following:
1. Burnt rear clutch.
2. Valve body malfunction.
3. Transmission overrunning clutch broken
4. Input shaft seal rings worn/damaged.
5. Front planetary failed/broken.
TransGo, Superior, Fairbanks, even B&M all make shift kits. Typically, they offer three stages. Stage 1 is street/RV/towing. Stage 2 is Street/strip. Stage 3 is full race/manual shift only. You sound like stage 1 is what you're looking for. Look for quality kits. Borg Warner is a step up from factory clutches. Red Eagle and Blue Plate are terms associated with quality frictions and hard parts. There are several vendors out there. I have dealt with these guys:
I list their site as an example of what all is available at a glance. Like I said, there are other sites. Lots of stuff on eBay. As with all things, you get what you pay for. Better parts hold up to hard use better.