I can't wear wool either, irritates my skin. I have quite a collection of cashmere all amassed on a budget.
Second hand cashmere is very reasonable. Look on ebay. Uniqlo do inexpensive cashmere but I haven't seen it in RL. H&M have some at the moment. Some of my fuzziest cashmere is from John Lewis and Gap. Cos is good quality too. J Crew cashmere is nice bit very fine knit IME.
Look out for sales and discount codes - there will be loads for all of the above brands between now and Christmas.
In terms of coats, padded coats are far warmer than wool.