Other preparation you can be doing now is paying visits to conservation labs. A graduate degree may not be so important if you want to work as an independent textile conservator, and don't care much about participating in some aspects of professional life. Still, having been through one of the programs is a good idea these days. I didn't do it, but I apprenticed more than fifteen years ago and things were quite different then. And graduate school does provide an element of networking that could be invaluable.
Money is a real problem in textile conservation.
Although relative salaries are much better than they were a decade ago, until you are established, you may have to choose between eating and paying rent.