Hello,
I hope I'm posting this in the right place, if not- I apologize.

I recently became inspired to try the Couch To 5k program.
I started a few days ago and I'm really enjoying it.
I get up and do it before work, and on the "rest" days I take a simple walk.

My issue is that today when I finished I got home and
felt awful. Really terrible. I had to sit down in the shower, I felt like I might
throw up. My limbs felt heavy and weird and the "about to pass out" feeling lasted for 45 minutes.

I ate some cottage cheese and that sorta helped, but I'm not sure if it was the food or simply the passing of time that helped me feel better.
Also, I'm not extremely out of shape- I've exercised at my highest weights and am generally an active person.
The only times I've ever passed out was when I was trying out the sauna at my gym. Apparently sauna's aren't a good choice for me.

So, was I possibly just too hot? Or should I have eaten first?
Any help or advice would be appreciated!

Talk to your doctor!
I've done some first aid work for some fun runs and marathons and this is not uncommon.
Sounds like dehydration, the body copes whilst it has adrenaline pumping but once the adrenaline subsides people collapse. It's water and electolytes you need to replace. Did cheese have salty flavour - that could have been enough.
Also with C25K programme - I've started recently also - and found some advice - take the Runninng/ jogging slowly to start off with - you should be able to talk as you run. My jogging is slower that my walking but my heart is still going faster.

I also think it could be dehydration. Also, how much did you eat before your run? Your question makes it seem like you ate nothing. If it was nothing or just something small like a piece of fruit perhaps eat more, and make it some carbs like oatmeal. Then you mention it is hot, so it could have been overheating. It could honestly be anyone of these things. It use to happen to me a lot when I use to workout non-regularly, for me it was overheating. If its that your face will be extremely red, you'll be really hot, your hot shower would have made it worst.

I personally cannot excersize at all without food in me. I get dizzy and weak after I wake up if I don't have something in me within an hour or so. I've tried to work out in the mornings, and I got the same "I'm literally going to die" feeling - and that was when I was very fit, at my lowest weight and working out frequently! We all have different bodies and needs. My best time to work out is at night