I am making assumptions here (I am sorry if this doesn’t apply to you, I am saying this in good faith and trying to help) but I think that a lot of people who have major issues with the intonation of a saxophone are often placing the mouthpiece too far away from the horn on the neck, then when they tune they involuntarily push too much their embouchure effectively bending the sound in tune and as they train themselves to that tension, the pitch of the lower notes drops lower (because to make them sound “ nice “ they relax the jaw) and the pitch of the higher notes become too high because from an already tense embouchure , as most people do, one tends to tense it even further when going up to higher notes especially if one tends to compensate a lack of breath support with excessive volume and air flow.
IF this is the case, then one needs to retrain oneself by pushing the mouthpiece further , consequently relaxing the embouchure. Of course in the beginning the whole process will be a mess but slowly you will retrain your body to a different embouchure and tuning.
If , on the other hand, this is not the case, there is always the possibility that something is wrong. Leaks are often the culprit because they make you push more to get the sound but they most probably make you play higher in pitch rather than lower, because you are pushing more, but you might have adjusted your medium pitch with a leaking horn and therefore would be in a situation like the one described before.
There is the chance that a horn is wildly out of tune. In general this is not the case of one horn in particular, unless someone has lowered the keys too much or cut the neck, but in general this belongs to a “ strain” of horns. Once I had a misadventure with a Super 20 alto made in USA (post Eastlake) about 1980. This horn would not play in tune. Other folks better than me couldn’t play in tune with it. I have tried a Silversonic of the same generation which couldn’t be played in tune by the shopkeeper nor could I play it in tune.
This shouldn’t be the case of a Mark VI. Quirks yes, wrong design no.
Which brings us to how would a new neck affect tuning. Unless your saxophone's neck was altered (shortened or lengthened) another neck wouldn’t really do too much to its tuning assuming it is of the correct length. The taper could vary, but not by much, but that would influence mostly the equilibrium of the various harmonics which make up the sound (varying the colour) another thing is , of course, the position of the octave pip which could throw registers even more away from each other (or closer) depending of its position.
In other words, changing neck could easily have no effect or even a negative effect, the odds are against the possibility that it will improve things.