Would longer axle to crown be ok? I have a Trek 820 2014 26 inch for which I had the lbs convert to rigid fork. The original suspension fork was a suntour m3030 which the manufacturer states has a 63mm travel for my bike size. I lost that fork so I have no idea what the length of the axle to crown of that when I’m seated. 

My current fork is actually a decent and durable but has about 375 mm axle to crown. I have my eye on a surly 1x1 in which the 100mm travel corrected axle to crown is 413mm. My fork has a trail that I don’t know the measurement of. So does this surly.

My issue is that my shoulders and neck get easily uncomfortable where I have to sit up on my bike(holding my grips with my fingertips) from time to time on only a 10 mile ride. Doing no hands is twitchy, very sensitive to movements so great care is needed when drinking on the go. I have a different bottle now though so I get to drink only during stops, or stop myself to drink, but still, the fact is maneuvering needs good focus.

Additionally, I have thinner tires now that accommodates 1.8 in tubes. I had no ride problems with the tires on the suspension fork, nor probably the rigid ones.

My question is will a higher front make the ride more comfortable for my shoulders and neck? Could the bike tolerate that change in height structurally?
You should not try to adjust your riding position by messing with the steering geometry. Riding position should be adjusted by replacing the stem with one that is longer or shorter of has more or less rise.

However, it sounds like you need a longer fork anyway. 375 mm axle-crown sounds far too short. I put a rigid fork on a 90's 26" wheel MTB that only had 40 or 50mm travel and the axle-crown was 420mm. I suspect you have steepened the head tube angle too much which is why the bike steering is too fast.

It's possible to make an estimate of the crown-axle measurement of the original fork by making some relative measurements on a high resolution image of the bike knowing the rim diameter is 559mm. The Trek archive page of the 2014 820 is here which provides just that. I estimate the axle-crown is is more like 460mm.