Couldn't edit my above post, so here it goes:

If I were you, I would do the following:

1) let Leo handle the dogs
2) get my ass to BKK airport, with a ticket to VTE, travel by train from KK to Hualamphong, take taxi to BKK, settle overstay at the airport and get to VTE by plane.

I understand trying to get to Nong Khai, might seem a lesser risk, but I am not sure if they will allow you to cross over to Laos with an extensive overstay, there is a good chance they tell you to go to the airport and leave that way, so you might as well do that in any case.

Going to KK immigration will get you to Canada, not Laos, that much is almost 100% certain.