| Wrangell has some REALLY good information there. Definitely has laid out some sensible training tips.
It sounds to me like you're trying to push it WAY too much. Running is kind of like when you first start swimming. If you push too hard you feel like you're getting nowhere, it hurts, and you get injured. If you relax, listen to your body, and push just hard enough to feel it, but not so hard it hurts, you'll see gains MUCH faster. Plus, you'll enjoy it.
Almost every triathlete has one sport that they don't feel confident in. Mine is the swim. I know I CAN swim, but I still panic at the start of the swim. Because of that, I put a lot of focus on my swim training. The tendency of new triathletes is to focus on what they do best. For you it sounds like that's biking. So it becomes a cycle (no pun intended) of doing better and better in your best sport, and still feeling inadequate in your "worst".
If it were me, I'd drop the second spin class. You're a biking fool anyhow, and that spin class is meeting that point of diminishing returns. In other words, you're not getting nearly as much benefit out of a second spin class at your current state of bike fitness as you used to. That's normal. It's because you're very "bike fit".
Instead, head OUTSIDE and work on your running. (Actually, I'd do the running first so you're fresh, then spin or bike. Biking does great things for "spinning out" your muscles after a run.) Dedicate that hour to your run. That doesn't mean you should go out for an hour and run every day. No way. You'll injure yourself. I'd start out small; ten minute warm up (light jog or brisk walk for 6 or so minutes, then some dynamic stretching), run at an RPE of 4-5 for 15-20 minutes, then 10 minute cool down (slow jog, brisk walk, then good session of static stretching). If you don't have time for all of that, make sure the warm up and cool down stays. They are very important.
Of course if you're going straight to spin class you can do your cooldown after that, but stretch WELL!
Shin splints are very normal. Almost all new runners get them. Your calves will develop faster than your little muscle in front (which I can never remember the name of), which pulls it and causes tightness. Ice ice Baby! Some great shin-splint exercises are:
1. Tap up and down with your heel on the floor. (I think I tapped constantly for 3 weeks straight when I had shin splints.)
2. Draw the alphabet in the air with your foot. It loosens, stretches, and actually builds those small muscles.
3. Point and flex. Point your toes out as far as they can go. Hold for ten. Then flex them up as high they will go. Hold for ten.
4. Take off your shoes, find a nice patch of grass that is dog-doo free. Walk on your toes for ten steps. Walk on your heels for ten steps.
These also help the ankle muscles. (And as I was typing them I found myself doing them, and it felt great.)
How are your shoes? Have you picked up a good quality pair of shoes? If you haven't, run your butt to a good running store. Not a mall store. But one that is stocked with actual runners, not teenagers saving for their first car. (Not that there's anything wrong with that.) They should watch you run and let you try on 4-6 pair of shoes and run in them. A good store will allow a 30 day return if they don't work out, and yes you can run outside in those 30 days. Their goal should be to get you in the right shoes, not make a single sale of the latest, greatest, shiny pair.  | 
