Hi Justin, you can do it but be gentle. Forceful stretching of small muscles such as those in your forearm when they are tensed up and full of lactic acid can place strain on the tendons and particularly the insertion points where they join the bone. Bend the fingers until you feel a mild stretch and then just give them several seconds to respond and relax, avoid pushing hard into the stretch, Im sure you can feel that this is painful! Be kind to your body!
Try shaking out with the arm held above your head and then hanging, alternate this to let gravity help clear the toxic blood and get fresh blood in there. Also remember climbing is catabolic and causes microtears in the muscle, these take time to heal so if you are sore the evening after a hard route or the next day you need to give your body time to heal before training again.
Warm ups- try massaging the entire forearm and hand gently before climbing as part of your warm up, I have suffered from the dreaded elbow for as long as I can remember, religiously warming up and massaging as well as some yoga/stretching afterwards and the next day are the only things that have kept the symptoms at bay. Additionally you may have inbalances in your shoulders or arms that can exacerbate this issue, get checked by a physio to identify any problems and sort them out.
PS theres also a good article on the massage techniques but cant find it on the web. I'll see if I can scan it and send it to you.