I've got one week overstay unfortunately. Will it be a problem to fly domestically to Udon Thani?

Airlines check ID, not overstay. But checking into a hotel might be, they do check for overstay and report your stay to immigration.

I'm not going to stay in a hotel, I would leave 7 am from Don Muang on my way to Vientienne and paying the fine. I do have a couple of days overstay from earlier. Will this be a problem?

No, they only look at your ID-page.

Why don't u just go Immgration and pay yr overstay ????

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You can check in for a domestic flight in Thailand with any photo ID. As long as it had your name and photo, your local library card would do it.

I always check in with my Thai driving license.

Thing is that if you get caught anywhere, on the street, in an airport etc. and they see your overstay you could be taken to jail first and that could mean not only pay your overstay but extra fees. As said, why not just go and pay it?

Why don't u just go Immgration and pay yr overstay ????

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Overstay has to be paid on exit of country or application for extension of stay (if qualified). Immigration would be required to detain if no exit made.