STDs are spread by fluid to fluid contact. So, if there was fluid in the shared underwear and it got into your body then yes, its a possibility. The only 100% way to no for sure is to get tested
Can You Get Chlamydia By Sharing Underwear?
If you don't wash your underwear, maybe
No, It's an STD, the 'T' meaning transmitted, which is the way it is spread.