What is a rash that looks and feels like razor burn and almost goes away until you get your period and then comes back?

If you shave then obviously this could be razor burn.

Look at changing your menstrual products. If you use commercial pads such as Always or Kotex this could be causing you problems, these are made from synthetics and plastics so prevent air flow to genitals which causes sweating as well as stops blood drying which can cause irritation. Commercial pads also contain hundreds of chemical ingredients, even if not scented, which can cause irritation.

Opt for cotton pads like Emerita or Natracare which are hypo-allergenic, or better still would be to use cloth pads like Lunapads or Gladrags, you could also use menstrual cups or softcups which are internal but unlike tampons there's no need to use pads at all.