My guess is that I need to move the cleat back on my right shoe. Does this sound correct?

Try loosening the front strap on your shoe first.

Yep.. Mine used to go numb and i loosened the straps and it helps..

I'll give it a try. The front strap on my shoe never struck me as tight, but I'll see what happens.

There are nerves that run between bones in the bottom of your feet (balls of your feet) that tend to compress while cycling and cause numbness in the toes.

Try using an insole that you have cut out the section where the space between the bones is.

my toes were falling asleep also... well actually they were going quite numb 30 minutes into a ride... i moved the cleat all the way back and started rocking super feet insoles (they have good arch support).. and i started making an active effort to put the pressure into the balls of my feet/arch rather than my big toe + lil toes.. the ball of your foot is directly below your big toe, before your arch.. in case you didn't know (i just learned this in yoga the other day lol)

Could be your shoes are too tight or too small.
A better insole can help, but it may not fit if your shoes are too small.

My feet used to go numb too. I moved the cleat back to lessen the pressure on my metatarsal heads and loosened the laces. Seems to have helped.
But what I think worked more, was to ease up on the pedal pressure from time to time.