Water heater plug

I doubt it as the water is what? 180F? Even if the coils were hanging in the empty tank for days with hot water in them, doubt they would fail. Just not hot enough.

BTW, most Class A RV's I'm familiar with the water system can withstand 20F weather for many days before you need worry. You didn't tell us the year or brand of RV you have but newer RVs, say from 2000 on, also have PEX plumbing that is much more tolerant of freezing then the older grey stuff.

So partially fill the onboard water tank at the first overnight stop you stay at, than fill your water heater tank...your system will be fine. Just do the regular...before bed, turn off the shore water, or turn off the pump if you're boondocking.

Motor aid should not get hot enough to do any damage.
While winterizing, I drain the water heater and then re-install the plug. I replace the plug if corners are rounded out.
I would not be concerned about temperatures just below freezing while traveling south. Not sure if your rig has heated water compartment. Mine has that feature and have traveled well below freezing with no problems.

I was wondering with motor aid heating the empty tank cause damage to the water heater.

I drain the tank and put the plug back in.
Why leave it out and let the bugs crawl in?

When in storage I always turn the plug in a couple threads, it keeps the small creatures out.
If traveling with an empty water heater I would install and tighten the plug so it doesn't get lost.

Loot at the ling range forecast for the area your traveling through. If its below about 30 overnights, then do not fill your water system. I assume your talking about 2 nights maximum? Fill the system your second morning and you should be OK. BTW leaving the plug in or out has no bearing on anything.