When I press any string at the first fret the notes sound correctly, but if I increase pressure they... choke out is the only phrase I can think of to describe it. I raised the action, which dealt with the problem, but I don't really like having the strings that high. Would adjusting the neck be the solution I'm looking for? I haven't delved into neck adjustments yet and am hesitant to do so on my own.

I'm having this problem with a Spector Legend 4 Custom - any folks with experience adjusting Spector necks want to point me in the right direction?
It sounds to me like you're pressing too hard. You only need to apply enough force to fret the string so that it will create the note. On most guitars, such as those with jumbo frets, pressing too hard will cause the note to go sharp. Applying the right amount of force is part of learning good technique.
You could have a go at adjusting the action, or maybe the truss rod but I'd attempt the action first as its easier to put the changes back quickly!

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Adjusting the action definitely solves the problem, its just that the strings are higher than I would like them to be. I tried adjusting my technique to cope for the choking but its not an ideal fix - maybe I should just deal with it until I can get a proper setup? But I like being able to take care of my instrument without relying on somebody at a shop who might know more than I do.

Maybe I'll put them way high up and try to get my Jamerson on...
A decision to adjust the neck shouldn't be made without an evaluation of the guitar. Most guitar makers post the procedures for checking the neck setup on their websites. Most of the time, it involves fretting a note at a certain string/fret and then checking the clearance between the string and another fret along the neck. Usually this is done in two locations along the neck. If the truss rod is out-of-adjustment, be aware of two things: The first, that it's quite possible to overtorque the rod and damage it, and that the adjustment usually takes a day or two to completely set in. So, you might adjust it a quarter turn today and see no improvement, however tomorrow you do see the change. By the same token, it's also possible to adjust it and see an immediate improvement, only to have it out again tomorrow or the next day. Another thing to consider is room humidity. Even though we're talking about an electric, they are still effected by humidity. I attempt to keep my guitar room at 50% RH and I see my guitars respond as the relative humidity fluctuates up and down from that point.
Yeah, and with winter setting in here the heating in my cabin fluctuating in all sorts of crazy ways. I'll take a gander at Spector website although I don't think I've seen anything there about neck adjustments on previous visits.

I'll ask around my campus, there might be a closet guitar adjuster.