Oh, I am so sorry you are going through this. I am a hair snob myself so I freak out when this happenes to me also. Just so you know low carb itself, will not cause hair loss but a the radical change in your diet can - because of loss of nutrition.
Just wondering what the rest of your diet is like? Are you getting enough protein?
When I was diagnosed a little over six months ago, I freaked and went crazy with the diet. I am a vegan so my diet is already pretty restrictive but yes, once I figure out the low carb part - that is a big dietary change and any time you drastically change your diet, your body will respond. Also a big stressful event will do it too. Geez, I had both! But the hair loss will often not manifest itself until many months after the events take place.
I noticed my hair thinning quite a bit in the last month and it freaked me out but when I think about the diet rollercoaster I have been on and the stress I have been through - I am not surprised.
This can be fixed if your problem is indeed either the thyroid, stress or diet. You can get your hair back - it will be slow process, growing back but you can.
I also have dry hair that never seems to grow in soft or fast - but mine is because I took Accutane a little over 10 years ago for bad acne and it shrinks your sebaceous glands - so it left my hair in a permamantly drier state. (if I had know it would be permanent I would never have taken it!) So now I have to deal the best way I can with the dry hair. I don’t even do anything to my hair - no color, chemicals, and I don’t even blow dry it!
I started a thread a few weeks back on here about diabetes and hair loss:
https://forum.tudiabetes.org/topics/does-diabetes-cause-hair-loss
As a former hair dresser (and one who has screwed up their hair many more times than I can count!), I can attest to the fact that stress and diet will cause excess hair loss. Most people spring back from this.
I like my hair long but it was getting so dull and dry the longer it got (thereby, dry hair is weaker and sheds and breaks more often) I just cut a couple inches off my hair (which I hated to do because it took me forever to even get it that long - it grows so slow!) Just cutting off the dry ends helped alot with the fallout.
Don’t forget your hair also goes through cycles were it will shed more - end of the summer (like right now) is a time when it can happen for many people). These hairs were on their way out to make room for new ones so dont’ fret too much if you notice this at certain times of the year - this is normal.
I starting myself back on some hair vitamins that i used to take years ago that worked well for me. I am also upping my protein and getting back into scalp massages (very helpful if you are stressed and have a tense scalp - get the nutrients going up to your follicules!). I hang my head over the end of the bed for about five minutes to get extra blood flowing to the scalp and other goofy tricks! (ok I know they sound crazy but they honestly worked for me in the past!)
I know how you feel so if you need any other hair advice, please let me know. 
I have totally demolished my hair in the past (several times!) and will a bit of extra effort, got it growing thicker and healthier than before.