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I was running an overnight print out of taulman's 910 nylon, and came down in the morning to a stringy mess that had wrapped around my e3d v6 hotend. There was nylon on the heater cartridge, and the hole for the thermistor looks like it got nylon inside of it.

I heated it up and cleaned the majority off, but now it will reach temperature for only a few seconds before it quickly starts dropping back to room temp. The target temp is still reading high. Has anyone experienced anything like this before? I'm trying to figure out if I should target the heater or the thermistor, or just replace both...

I doubt the nylon has anything to do with it in the way you suggest. What seems most likely is that you accidentally loosened up the heater connection as you were removing the nylon glob.

The thermistor is probably fine if it's still reading plausible temperatures.