Whatever you do, don't ignore it. Get treatment for it, rest, stretch, whatever--but don't work through the pain. You'll end up causing permanent damage from a buildup of scar tissue.
I fought it for a few years in my weak-hand elbow (from typing, not shooting, but it certainly did prevent weak-hand shooting too). Did the cortizone shots, physical therapy, etc and finally had to have surgery last fall. I'm still only about 80% in that arm, but it's slowly getting better.
For a stretch, hold your arm straight out in front of you. Pull back on your fingers so your wrist bends up and back, hold for 10s, repeat 3x. Do the same pulling down too. There are a few others you can probably google on too.
After using it a lot, and before bed, do a 2 minute ice massage on the tendons near the elbow (wiggle your fingers, see where it moves in front of your elbow--right there). Freeze some water in a styrofoam cup, peel the cup back to expose some ice and rub.
As soon as you can do it without hurting yourself further, do some exercises to build up strength. Some I can think of, I do about 3 sets of 10 of each of these every other day (except for the string one). I stopped these for a few weeks once and the pain started to return, so I plan to keep them up indefinitely:
- elbow at side, forearm straight forward, palm down. You could even do this on the arm of a chair. Grip a weight (just a couple pounds, a can of beans, whatever you can manage) and lift your wrist.
- rotate arm so palm is up, repeat above.
- regular curls
- elbow straight up, arm bent so you're holding the weight just above/behind your shoulder. Straighten your arm (straight up), sort of a reverse curl.
- get a heavy object a foot or two long. A flashlight, hammer, pipe wrench, stick, whatever you can find and manage without hurting yourself. Grab the end of the object, arm out in front of you, and rotate your wrist side to side so the object swings vertically through 180 degrees.
- you'll have to make a tool for this one, but it really works. Take a stick like a broom handle about 18" long, drill a hole through the center and tie a string or light rope. When holding it in front of you the string should just reach the ground. Tie a weight on the end of the string (I'm up to 3lbs now). Hold it out in front of you and turn it (using both hands) to winch up the weight, and then back down.
- Repeat above spooling the string the opposite direction as you did previously.