Perhaps raising the deck up a notch will minimize the accumulation of dead grass? Has this happened before or is this a new occurrence?

A different blade won't eliminate the windrow of cut grass. Your mower is simply processing too much grass at one time. Try cutting a narrower swath, walking a bit slower and most importantly mowing more frequently.

Unrelated to the grass accumulation problem, but I have a Toro model # 20007 and bought a newer Toro 22" Atomic mulching blade. Can I use this blade with my model even though it's not the the part listed for my mower?

I don't know, but they come with Atomic blades. See how it fits and chack for clearance all around before starting.