I want to join a 1" wide by 1/4" thick by 8 feet long strip at a right angle against a board that is 3" wide by 1/4" thick by 8 feet long, creating sort of a "T" profile.

I have planed the strip edge virtually flat and straight. But my board is not perfectly flat - I have no planer machine and my hand plane tool is not good for that 8 foot by 4" wide board. I'll use a long sand block to smooth out the small ridges but the board is pretty much as it came out from my table saw - with some hills and valleys that sanding will not fix.

The question is two-fold. First, would I be better off using wood glue (Titebond III) or epoxy?. And second, would thickening the chosen adhesive with wood dust collected from my table saw be fine?

This needs to be structurally as strong as the wood will be if possible - I don't want it to break at the glue joints. Would that be possible without a perfectly tight joining area?

Thanks!