I ran the new 10-3 wire yesterday and hooked it up in the panel box. Dryer started right up. Thanks for the help!

I opened the receptacle and it has NO neutral wire connected. It just had some 10-2 wire coming in with white and black on H1 and H2 and the bare on G. The silver N slot was just tightened down with nothing in it. When we got out previous dryer 2 years ago, it was a three prong. The hookup at our new house was a 4 prong, so I got a 4 prong pigtail and hooked the dryer up. This new dryer actually requires all four, so that must be why it's not working. I'm going to just re-run the wire with some 10-3 and see what happens. I'm hoping it works. I've learned a decent bit today. :)

Ok, so I just took the receptacle cover off and saw this: There is a regular white/black/bare wire going into the 4 prong receptacle, with white going to H1, black going to H2, and bare going to ground. There is no wire hooked up to the N slot.

Those are the readings with the dryer unplugged.

Also, I just went and measured again to be sure, and I'm still getting 240v H-H, 120v H-G, but now I'm getting only 10v H-N.

N-G measures at 0.1v... fluctuating (if I wiggle the probes) up to 0.7v.