TL;DR version at the end. I try to post all relevant details.

If you read my intro you know that I know.... not all that much. For the past two years I've been using Asics Gel Fortes (model t310n according to the tongue label). These were the shoes I was made to buy in Basic Training. I knew, and know, nothing about them other than I had to buy them and I was going to run in them quite a bit.

On Monday I went to the local running supply store to replace my quite worn out shoes. I told them I was having numbness issues in my left foot after 3.5 - 4 miles and tingling in my right instep. I was told the numbness was probably just because I run on the treadmill but that combined with the instep tingling could be because my Gel Fortes were 'stability' shoes with an extra stability pillar built into them. After watching me walk and jog in the store she said it didn't look like I needed a stability shoe and brought out three neutral shoes to try that supposedly fit my demonstrated running style. I chose the Brooks Launch 2's as they were most comfortable and felt the most secure on my feet. These shoes are INCREDIBLY comfortable to me while walking around.

Monday night I tried running in them on the treadmill. Initially I felt like my feet were much lighter, maybe bouncier, and that running faster was much easier. After three quarters of a mile my ankles and the muscles wrapping around the outside of my shins (the tibialis anterior and the extensor digitorum longus according to this image) felt extremely tired, and by one complete mile I had to stop. My muscles were so exhausted it was getting dangerous, as sometimes my foot (not my legs) simply wouldn't lift high enough, my toes or forefoot would drag, and I'd almost stumble. Even while standing still trying to relax, the muscles were so tight they felt completely rock solid. At that point I didn't even want to walk. After some stretching I walked around the gym trying to find anyone who identified as a runner to help me. No luck, but standing and stretching for around ten minutes had everything feeling basically normal. I tried again and had the exact same results. After the second attempt I opted to use an elliptical for the first time, which did not cause any issues.

It feels like my feet are making initial contact with the ground about an inch or two more forward, toward my toes, than they used to. It's my assumption this is causing the muscles that push/pull my foot away/toward my shin to work double time, on the landing and when pushing off.

I have a PT test on Saturday, so I have returned to using my old shoes until at least after that.

TL;DR:

Going from Asics Gel Forte's to Brooks Launch 2's, my outer shin muscles are at the point of complete exhaustion 1 mile into my normal runs. Symptoms go away in about ten minutes if I stretch and let the muscles rest but return at the same point if I continue running. Elliptical use causes no symptoms. I think the Launch 2's are causing my feet to land further toward my toes.

What is happening, and what do you recommend? As always, thank you for your help.

I agree with what runharriet had to say.  It is most likely due to a significant change in shoes so you need to use the old shoes and slowly ease into the new shoes.

I would also suggest returning to your local running store.  I would try and get a different employee to work with you - each has different expertise.  Explain to them what you encountered.  They might be able to determine a better shoe for you or confirm that a break-in period is needed.  Most running stores accept returns within a few weeks as long as the shoes are in reasonable condition.

Echoing what everyone else has suggested, perhaps trying something like the Brook's Ravenna 6, to split the difference between your previous shoe and the Launch 2, then eventually pulling the Launch into rotation for shorter runs and increasing the distance you can tolerate in the Launch.  When I first started running I was in relatively beefy, moderate stability shoes.  Over the years I gradually transitioned from those (when the degree of medial control started causing issues with my increasingly strong legs and feet) to a lighter stability shoe, then light stability/lower drop.  Nowadays I'm in the Saucony Kinvara, which is a neutral, lower drop shoe.  But there's no way my legs/feet/ankles/hips/knees would have tolerated that drastic change all at once.  It took months to work from one type of shoe to the next.

This used to happen to me when I was a new runner but it had nothing to do with the shoes.  I was a bike racer for 20 years and when I switched sports I would get the same condition you mention.  For me, it was just the different muscle usage between sports.  The main thing that helped, other that just putting in more miles, was doing toe raises and calf raises (I still do them a couple times a week).  Those exercises helped strengthen the muscles around that shin area and the pain never returned.