Long story short, there is too much relief on my 335, but before tightening I need to replace the nut.

Should I do this at full tension, with loosened strings, or should I remove the strings entirely?

Quote by Tony Done

The difference would be measurable in microns, if at all, and would likely be the reverse. - Taking the strings off might expose less threads because the wood is relaxed.

just the opposite, there would be more threads because there is tension on the neck
Tightening the truss rod should be done with the strings loose.  If I were changing the hex nut on a truss rod I would do it with the strings loose.  There is no reason to loosen the strings to change the nut other than the fact that you have to loosen the strings anyway before tightening it  down all the way.  Kentrock is correct that the string tension will give you a little more of the thread to work with so if you are worried about having enough thread to get the nut started then keep the strings tight to get it started but as soon as the hex nut is starting to pull the neck straight you should still loosen the strings before tightening the truss rod.
Not taking any online orders.