'02 Avalon Not Going Into Any Gear, After It Just Did.. ???

I just picked up an '02 Avalon w/ 277K mls. At first it would only go into Reverse. Drive, only 1 time felt like it wanted to engage but did not, and that was it. Days apart attempts, no Drive... (shifter position made no difference.)Reverse continued to engage normally. After sitting for a couple weeks, with motor 'warmed' but not at normal running temp., surprisingly Drive 'did' engage, and felt firm/normal. My driveway is a slight uphill grade. It traveled approx. 40+ ft., I shifted to neutral and coasted back. I did this twice. (I did NOT gun the motor like an idiot would.) Best I can describe it is.. Drive engaged with authority/Like it was supposed to.. it felt 'normal'. The 2nd time I did not coast back, I put into Reverse to back down drive. Felt normal. As 1st time, Full Stop. 3rd time.. No longer went into ANY gear. Is this the Module, the Solenoids, or tranny is bad? As firm as it felt going up the drive, I'm guessing it's not the bands, but at 277K mls? But it felt firmly engaged, and then, next second, (after going to reverse) nothing at all. (???)