Sounds like taking an L to me. I'd rather fix the problem than mask it with dumbbells.
If your arm doesn't hurt or feel tight at the house, it's almost definitely a form thing that's causing your discomfort.
You prolly need to change your setup. Your elbows should at roughly a 45° angle at the bottom of the press. Also make sure you're not benching with a completely flat back, as that will change the angle at which the bar path travels, but not for the better.
Also might wanna make sure you're doing enough pulling movements (back, biceps) for elbow and shoulder stability, and stretching after you lift (save the static stretches for after lifting; before you lift it should be dynamic movements/warm-ups only).