I was doing laundry in the first trimester too. Using a washing machine is totally fine, you're just pressing buttons and maybe carrying a small basket 
The main concern is heavy lifting or scrubbing by hand, which can leave you exhausted. At 6 weeks you might feel tired or nauseous so just pace yourself and let your partner or family help if you can. Listen to your body; if something feels like too much, take a break. Folding tiny baby clothes can even be a sweet way to daydream about your little one!