Alright, let me start by saying that my mountain bike feels great when I ride, so I don't think this is a fitting issue.

When I ride for any length of time I've noticed that my hands start to get a tingley. Its normally nothing I can't fix by stretching out my hands and just setting the bottoms of my palms on the handlebars til my hands stop tingling.

Well, thats how I fix the right hand.

The left hand however will keep tingling, sometimes taking 2+ days to stop. This is particularly bad in the pinky and ring fingers, not so much the others. Worse, when I hit a significant bump, or something that causes the frame to jolt, I occasionally feel a shock run from my wrist up my arm. This is only in my left arm.

Any idea what's causing this? Any idea how it can be stopped?

Flat bars are causing that. You're keeping your hands in an unnatural pronated position for too long, and the road shock only exacerbates the sensation. Try out some bar-ends and see if that helps. Having 2-3 different hand positions that you can change up from time to time can really make all the difference in riding comfort.

Also, learn to grip the handlebars with less force. If you use a death grip, ease up and use a firm, but gentle grip instead.

This has come up before, recently. Try searching for wrist numbness, or tingling, or ulnar nerve. The bottom line is that your body is protesting being in an unnatural position. Don't ignore the problem!

It may not be just your handlebar type; your bars could be too far from from your seat, and/or too low in relation to your seat. Since your hands are suffering unequal effects, either your bike or your body may be asymmetrical.

Gloves with padding in the wrong place can also be a problem. I like Specialized Body Geometry gloves, but a guy working at a local bike shop finds that Specialized Body Geometry gloves cause discomfort for him, because the edge of the rather prominent padding is right over a nerve.

Flat bars make my hands fall asleep.
I found great relief with Mary bars and ergon grips.

https://www.bikeforums.net/utility-cycling/422506-alternative-handlebars.html
^^^
A good read about handlebars

The Ergon grips sure helped me....

Bars are too low. You need to raise the bars so you are not putting so much pressure on them. As you get in better shape you can lower them to the desired height.

I was having the same problem on my flat-bar mountain bike, and I just installed trekking bars.



Because of the way the original brake/shifters mount, it worked out best if I installed the bars "backwards" - i.e: with the open end forward. I couldn't get the brackets to slide around the curves of the bar, and there probably wasn't enough straight bar there to mount them properly anyway. One of the benefits is that the position is a little more stretched out - the reach was too short for me with the factory bars, and if I stood up in the pedals I felt like I was going to do a somersault over the bars.

You're crushing your median nerve. Runs from your elbow to your finger tips. Your "crazy bone." Get some thicker palm cushioned gloves.
I've been through it.

Thanks for the advice guys. I'm gonna take a trip down to the LBS and see if they'll check my riding position and have some advice/suggestions for me. They've always been great at helping me so I don't think they'll mind. Thanks again!

Same here. I use gloves and Ergon grips and haven't had any numbness at all, even with rides as long as 5 hours, wheras previously only 2 hours was enough to cause problems.