My current passport has been filled up with tourist visa's and even a red stamp, as far as I can see I've no option but to get a new passport or leave the country. If I get a new passport here in Bangkok, would I have any troubles applying for a tourist visa in, say, Vientianne? I'm sure they'd like to see my old passport and may refuse me when seeing red stamp etc.

No problem to apply for a new tourist visa. They will not see any in your new passport. They will not ask you to show your old passport.