Thank you so much for your help.
I am very worried now because there's so little options

. Last year I got a quotation with a local shipping company and they quoted via Lufthansa. About forking out good money, it's very true because they quoted about 2500 euros.
I contacted them again to see if they still offer the service, just in case I don't manage otherwise.
At least ITA having Chicago as a destination, is a bit of a relieve. It is a long drive to Tulsa and back, but it's only once and will do anything to get my dogs with me.
All US airlines allowing pets in the cabin have no weight restrictions, just carrier size restrictions.But none allows more than one pet carrier per person, so flying domestically is out if you are on your own.
I always travelled from EU via United, but sometimes operated by Lufthansa or Swiss Airlines. Does it mean the same rule applies? i.e cannot have two dogs and one person?