Can I leave Thailand using both my old and new passport at the Mukdahan border?

You have to be stamped out in the same passport you entered on if you don't transfer the stamp, and there is another detail you are forgetting. When you applied for, and received, your new passport, your old passport was notched, and made invalid, so you can not use it to leave the country, and you can't leave on the new passport as it has no entry stamp in it, and your old passport is no longer valid, so you must get your entry visa transferred into your new passport. You also can not enter any other country if you don't have an exit stamp from another country. Departure immigration can not accept an invalid passport that has been voided, no matter what letter you have from your embassy. They do not offer the service of transfering your stamp.