Try to use your arms as hooks and think about pulling down with your elbows not with your hands. Lighten the weight up till you feel your back doing the work(swallow your ego) and build up from there.
Great suggestion and try a false grip no thumb as well it does a lot of taking the upper arm out of it
I second both suggestions.
USNS’s suggestion of changing up exercises, while not necessarily wrong (rows are great exercises), doesn’t really address the problem. Like Scott and Phil alluded to, your problem is that you’re pulling with your arms rather than with your back. This is most likely due to being a naturally arm dominant puller (meaning that basically you have better neuromuscular control/connections to your arm flexors than you do to your back muscles).
Now, as far as getting sore in your biceps while military pressing, that is strange indeed. What variation of military press are you performing?
Also, they’re not a back exercise, they’re a shoulder exercise. I’d suggest using the same method as suggested above and go back to square one with the military press and work on really just contracting your shoulder muscles against the resistance. Another tip is to once again try pressing leading with your elbows (meaning that you are trying to lift your elbows rather than to just focus on pressing the bar).
Both of those tips should allow you to get better at recruiting your delts and consequently improve the effectiveness of the exercise.
Hope this helps.