Itchy skin isn’t a thing for me but being cold all the time definitely is.
I had/have severe anemia, always itchy, always tired. I did lose weight, and I was super pale and always craved sonic ice. I also would almost pass out any time I ate because (according to doctor) the blood rushes to your stomach to digest food and since i had almost no iron at all it was using what little I had to process food. I ended up needing 5 IV infusions, and after 4 months I need additional infusions.
Itchy, dry skin and feeling cold can definitely be symptoms of anemia. I'm not sure if losing weight is also related.