What's going on in there

My first batch is in bottles about a week. I turn them over a couple of times and I see a little foam in the neck. Is this mean that it’s carbonating we’ll or is it just wishfully thinking. I’m very anxious to try one but going to wait 2 more weeks

That foam just means you’ve turned them over; but I’m sure it’s carbonating. I bottled a batch last week too, (Assaazin) I had about 6oz left over, which I put in a 16 oz water bottle as a carbonation meter. (Fill with your flat beer, then crush-out most of the air and seal it up.)
After a week, the bottle is just starting to un-crush. By next week I’d expect it to be about uncrushed, and after 3 weeks totally uncrushed and getting firm. That’s about the time to START thinking about putting a sampler in the fridge.

Your beer has gotten to a certain limited amount of carbonation, but you need to give it more time to finish the job. I consider 3 weeks to be the bare minimum amount of time to wait before sampling. Some people will tell you that you can crack one open in 10 days, but that’s pretty optimistic, in my experience. At that point, you may get an acceptable amount of carbonation, but you have to remember that the beer is undergoing a secondary fermentation in the bottle, so give it time for everything to settle into place. Patience is key.

Next batch I’m going to bottle some to a plastic bottle so I can monitor it.

Assuming sufficient, healthy yeast and an ambient temp in the 70s, you should have complete carbonation in a week.

The best I can due this time of year without heating the room is between 62 and 65

Then expect three weeks minimum carbonation time. The yeast know their job.

True, but it takes a bit of additional time for the yeast to settle out to the bottom, and for full flavor development. Personally, I wouldn’t consider the beer ready to drink in less than two weeks, and longer is typically better. But the exact schedule is going to vary from beer to beer.

Ill tell you what happened in there. I couldn’t wait any longer. This is my first home brew and also I ran out of beer. I couldn’t stand those guys staring at me, so after 2 weeks I opened one. Boy was I happy when I heard the sound, you know the one. Beautiful brown with a nice foamy head. It tasted very good too. Of course I will let it age a little and plan on putting some aside but I know my limitations so I will be sneaking a few more.

Congratulations on making beer!