Just an update on my attempt at repairing my gas dryer. After rebuilting the wiring harness around the valve coils, installing the new igniter, and reassembling the dryer that I've torn apart; I fired it up: there were flames in the burner and heat in the exhaust pipe!
I've been backtracking my fixes and replicating the issue: I believe I've turned the dial pointer on the AUTO setting too far (between "Less Dry" & "Off"). This probably knocked the dial outside of its operating range, causing it to stick (which sometimes happen on my little oven). The dryer overheated after running for 3hrs and possibly damaged part of the harness, shorting it somewhere. As a result, it never heated up afterwards; even after replacing the valve coils. Worse yet, the next morning, I found the the harness around the valve had melted badly and the wires from the igniter burnt up beyond use (I'd left the dryer plugged in overnight).
Now that it's back up and running, I'm able to successfully get the AUTO settings to operate normally by keeping the pointer within the "Very Dry to Less Dry" range; however, upon testing, I've found that the manual TIMER setting is much more consistent and haven't failed. So I'll likely stick to this rather than using any of the AUTO settings, just to be safe. Anyway, I appreciate everybody's contribution...not only saving the dryer, but more importantly, I've attained a new skill!