Are your gloves snug at all? Make sure they are loose enough- try undoing the velcro mid ride or even taking one off & see if that helps at all. Also make sure your handle bars are the right width, so there isn't any pinching happening in the neck/spine/shoulders that could contribute. (Bike fit if you haven't already)

I have tendonitis too... wrists are particularly bad. There is only one hand position I can ride in, all others cause pain or numbing - and that's on the outside of the bar, either on or just under the hoods. Thumbs are up, wrists neutral. I can't hold a flat bar, nor can I do push ups due to the wrist & hand pain it causes.

Good luck - definately follow up w/ the doc to make sure nothing else is going on.