Pictures would help.
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Yes, the NEC says you can run a separate grounding conductor, NEC 2014 250.130(C) HOWEVER you are not allowed to run regular (THHN) wire unless it's in a raceway (protected) So it makes no sense at all in this type of setting. You're much better off putting in new Romex to update the system entirely. You are confused on the support requirements. Within 36" of the box refers to conduit not cables, they are usually 12" from box under most circumstances, HOWEVER if you are "fishing" into finished walls the code allows you to put it in without support. NEC 2014 334.30 (B) (1)
Just wanted to clarify this so nobody gets the wrong information. I think calling a licensed electrician for a quote is a great idea, in fact get a few quotes.
get the panel upgraded. (pull a permit, we didn't once and got screwed by a licensed electrician - fail on his part to do it right)
Put some nice large junction boxes in the attic/basement where accessible.
Start running new romex to each outlet either from attic down or basement up, wherever the old stuff was. Use the old as a pull cable for the new stuff.
We did this on my house, and every weekend or so, run another new romex. Takes time, but the house is all new now wiring. Trying to fix up the grounds was as much work as just running new stuff.
Doubt I need a new panel. A room was built onto the back of the house years ago so the panel is pretty new and has plenty of space. My original plan was just to run new romex to each outlet and try and do a different room each weekend, exactly as suggested.