Can a saddle influence a position to an extent that it's just not for you?
Yes. Also, knee pain almost ALWAYS manifests itself 'after' the ride rather than during unless something is seriously wrong!
This sounds very similar to an issue I had a few years ago. It was only really a lot of trial and error and ultimately a bike whisperer bike fit that sorted it.
In my case I was using a specialized power. At 143mm it 'should' have been exactly the right width for me and was 'on paper' the same width as its predecessor which had been fine for years.
However, after a long hilly ride in Wales I started to get exactly as you describe, like shin splints. I also had unusually sore quads which I just put down to riding fixed up hills etc.
I went to a physio who even suggested I perhaps should stop riding fixed if I was prone to injury! The horror!!!
So upshot was, after seeing some photo and video of me riding it became apparent that I was only using about 50% of the saddle, the front 50% as the rear profile was actually much too wide.
What this was doing was shoving me almost a couple of inches too far forward. I admit I should have noticed the change but as it was a new saddle I just did the math, set it up and rode it.
I've since changed to a narrower but still noseless saddle with a profile that suits me much better and have no aches at all.
I do long hilly club rides weekly totally pain free these days :-)
Can't guarantee you have the same problem but could be worth looking into...