I have a tablecloth Gemach. Can they be used for both Milchig and Fleishig after being washed in between uses?

Yes. It should first be rinsed in a cold or lukewarm cycle and then washed on a hot cycle.

Sources:

רא”ש ע”ז פ”ה. מג”א סי’ תמב, סקי”ט. רמ”א יו”ד סי’ קלח, ס”ח. ש”ך שם סק”י. אג”מ או”ח ח”ד סי’ קו.