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After removing most of my honey supers a few weeks ago, each of my hives were left with a medium super of partially-capped frames that they hadn't quite finished yet. And no, I did not test moisture levels...

Question: what should I do with these frames/supers now, since they look the same now as a few weeks ago (they are above queen excluders)? Pull them off, stick them in the freezer and return during the fall flow, leave them on and let the bees utilize this food, or something else?

Any of the things you suggest would be reasonable steps to take. I tend to leave the partials on the hives as additional feed if the bees need it, and a start toward winter supplies if they don't have an immediate need. I don't count on a significant fall flow--it may or may not happen. If some hives need less space, I may move a super from one of those hives to one that can use the feed/space. The more partially filled comb I can safely leave on for bees to take care of, the better I like it. Open, drawn comb is easier to store and keep safe off the hive than is a frame with partial honey/nectar.

So I did buy a $25 refractometer, and was excited that even the uncapped frames were reading an average of 16.5% moisture! I pulled all the supers off, replaced with a super of blank foundation, and put on the feeders to see if there's any chance of getting them to work and draw out anything...

Each hive is going through 1 gal of 1:1 syrup per day! Should I keep the feeders continuously full for a period of time, or let them empty and only refill every week or so? Again, the goal is to try and get some comb drawn.

I keep them full to keep them working. I started a new comb drawing operation a few days ago, and they are also taking about a gallon a day of 1:1. After they go through about 5 gallons, I'll check to see if they are drawing comb, and if so, how far along they have gotten. But it will take a lot more than the 5 gallons to get a super drawn.