without turning this into an unfriendly argument, like on Thai Visa, the USA does indeed let you travel after you have ordered a new passport. Tod Daniels, resident visa guru and admin of this group, even acknowledged that earlier in this post.
Perhaps your country does not but USA even allows ordinary citizens to hold two USA passports at the same time so they can travel to Israel and other Middle East countries. So End Of.
There is a VERY BIG difference that you just put in your post. You said your passport was lost or stolen. So of course, because of terrorism security concerns, your passport was immediately cancelled.
You can always buy a new US passport without it being lost or stolen, does not matter the age. No need to damage it by washing it either. You do not even have to provide a reason. Just pay the money and new passport.
Tod the US Embassy in Bangkok does outreach in cities around Thailand. You can apply for your new passport and keep your old one. They do not cancel it yet. You can then go to Bangkok to get it or they will bring it to you on the next Outreach visit. You show them the old passport and they cancel it then and give you the new passport. If you do not collect the new passport they will subsequently cancel the new passport.
You do not have to wait till the last minute to get the extension. You could have gotten it before heading to Koch Phangan (for future reference). Or you could get it anytime after. You do not lose any days by doing an extension early. Days are tacked on at the end of the period.