Toilets, Sinks, Showers, Tubs and Disposals - broken garbage disposal or clog past J trap?

I've got a double kitchen sink. I came home today to a clogged sink. If I use a plunger to clear it on the disposal side sink, water wants to go to the other sink. If I use the garbage disposal it seems to drain, then the water comes back up again.
I removed a couple of the pipe connections under the sink to drain the water into a tub. The J trap is definitely not clogged. I drain all the dirty water out, then put the connections back together, then run water and it drains fine.
I turn on the disposal with water running and it clogs again. Will not drain.
The disposal does turn.
I had the guy on the phone at Do-it Center tell me that it's not the disposal and that it sounded like a clog past the J trap.
What do the experts here think?
Appreciate any help.
Thanks,

rhs

Ok, I take that back. If I just run water (disposal off) into either side of the double sink, it will not drain. J trap feels ok, not clogged.
It is possible that it is clogged past the J trap? If so what's the best remedy?
There's about a foot and a half of pipe there before it exits the house.

Appreciate the help.

rhs

Remove the trap and either buy or rent a drain snake. You can get one at a home center for about $10. Has 15 to 25 feet of cable on/in it. Run the snake down the drain watching to make sure it goes down and not up (into vent) after it goes into the wall. Disposals push water at a high rate and thats why the water backs up when using it. Good luck.