You are on point with this tensing up part, I press frets harder, my thum pushes harder against the neck and everything is sloppy... I press harder because I tend to hear my strings buzz, but that might be because of the action and the fact that I get tense when playing.
You have to stop this sort of thing. Relaxation is key. Fretting strings, even with high action, does not require a lot of force. The same goes for the right hand, whether using fingers or a pick, you don't need to pound on it. The tone actually gets better the lighter you play, it chokes up when you use too much force. Less force also makes it easier to play faster. When you feel yourself getting tense, start taking deep breaths to relax yourself.
If your hand is cramping STOP PLAYING. Try some stretching exercises, I'm sure you can google some up quickly enough. When I moved from play bass guitar to double bass I had to do exercises before playing (like before every set of a gig for five minutes!) for months before my muscles were used to it.
Also, don't worry about speed. Practice difficult parts below the real tempo until they are totally effortless before even trying to play at full speed. Remember this: if you can't play it slowly you will never be able to play it fast.