Not a vegan but I track my food for my sport.

I'm assuming all these are new symptoms (I'm not a doctor btw, so this is just a guess)

Hair thinning: the usual suspects are stress, male pattern baldness, menopause, but a suddenly very bad diet might do it after a few weeks/months

More tired: Could be related to a lack of complete nutrition which would be typical of a vegan diet if you aren't making specific choices to avoid deficiency (B vitamins, some amino acids, etc.)

But it could also be that you are eating a lot fewer calories (vegan food is usually less calorie dense and potentially more filling, so you could be eating fewer calories without knowing), which would make you feel more lethargic

Going to the toilet more is likely just greater fluid/fiber intake

Being more cold could be because you are eating less (happens to me on weight cuts)

Dry skin could be, but if it's winter where you are that could be it too.

Probably best to ask yourself how long you've been doing the diet for, and if you are losing body weight on average week to week. If you can weigh yourself every morning after going to the bathroom and plot your weight versus time that would give you a good idea. Plus you can estimate your calorie deficit from that graph. The rule of thumb is 3500 calories per pound of body weight lost, but this isn't completely true. It still gives a good idea though.

Anything more than 500kCal per day deficit ( around a pound lost per week) and that could explain everything, and perhaps provide evidence of a nutrient deficiency (if you are eating fewer calories, and from different sources than before, you will also be consuming fewer nutrients).

For micro nutrients talk to a dr or dietitian

For macro nutrients (protein, fat, carbs), unless you are training in a gym, try and get 50-100 grams of protein a day, 60+ grams of fat, and 200 grams of carbs.

That should cover everything diet related :)

Anything else and you'll need a professional