Can you confirm that the gear selector is positively engaging the detents for both forward and reverse when you do it manually? Seems odd that you arent getting either gear.
If it was a damper plate, typically you would hear a lot of rattling or sounds at idle or during a rev if something was broken but it could be that the splined section has come free or something odd. I had a couple of springs let go on mine and it was quite the noise.
Either way, to check the damper plate you have to pull the trans/bellhousing off so it is probably not a bad idea to get the trans refreshed while its out.
1985 Supra Comp TS6M
351 Windsor - 240 HP
Velvet Drive - 1:1
Fully Rebuilt