Originally Posted by Wildwood

I'll take a few shots =

Gloves? ~ Yes
Gel bar tape? or anything similar. ~ 2.7mm Lizard Skin
Are you right handed? ~ Left handed!
Controlling speed with mostly rear brake? ~ No, mostly front
Supporting too much upper body with that stronger right side? ~ It feels the same to both sides
Broken collar bone either side to change angle/shape of right or left shoulder? ~ Never, thankfully
Do you ever sit-up and 'shake it out' to relieve pressure points? ~ 10mi seems a long time to be holding onto the hoods without a change of position. ~ I change positions fairly often. When I feel the tingling I sit up, let my right arm drop by my side and wiggle my hand around...
Has this pain occurred in any other activity? ~ No
History of nerve issues? ~ No

Fitter is correct, brake hoods must be parallel and 'forward-facing' the same direction for good feeling. Unless your arms are slightly different lengths.

edit: As a new rider, learn to relax on the bike. Really relax. Like, visualize your Happy Place when you start the ride. Focus on breathing, cadence, and moving around on the saddle, bars and pedals as often as needed to relieve pressure points. Stretching on the move is a good thing for other muscles, too.