I sometimes do a little drill where another guy tries to use my face as a double end bag. He tosses on gloves and just jabs to his heart's content, and I just bop around trying not to get busted. Other than helping head movement/blocking, you're guaranteed to be hit when someone's throwing speed jabs in volume without any fear of you throwing a counterstrike.
Although someone might disagree, I think workin' the heavy bag helps the problem you've described too. You said you have problems just letting your hands go, because you tense up. So punch the bag till you're about dead exhausted, til your shoulders and arms are on fire, till your lungs ache, and then throw some more punches. Gets you in that mindset of pain is pain, but regardless of how much anything hurts - eyes forward, chin down, hands up, and swing.
No need for me to explain the obvious experience of sparring and getting hit any further than it's already been explained by others.