I think some of it, Coolpink, is that when we are intent on getting the intonation and timing correct, there is a tendency to bring the fingers down much harder than they need to be for getting the note to play.  Then you end up needing more pressure from the thumb and those two things sort of increase together until one is holding the neck so tight that shifting position and etc is impaired. 

So far as the "lesser" songs, well, some of the simplest melodies are very memorable and pretty in their own way. 

A thing I have found a bit helpful I to start practice with maybe 3 minutes of just playing long slow notes on the open strings while walking around a bit.  Since the left hand fingers don't have to actually do anything, it lets my bow hand "wake up" a bit and remember the angles for getting good notes on the string and keeping the bow moving straight.  While I am doing that, I try to focus on the sound of the notes and the feel of playing and just sort of get lost in it for those few minutes. 

Then I do a few minutes on each string, playing a melody that can be done on just one string, to warm up the left hand and actually "be playing something".  That can be a bit more fun and refreshes my fingers' memory on the basic intonation. 

Then I will do whatever scales I'm working on that day, and whip through an easy song or two for fun and then take a short break and then finally get to some "real practice".

By then I have already had to remind myself a few times to relax, to keep the bow moving straight, and etc, so I tend to do less of those mistakes after taking the time for warming up a bit.  I think it also gives the brain a chance to shift to "artist/musician" mode from usual state.