Just in case you have to go to that Children's Hospital, start keeping notes now. Note how many ounces of fluids they're taking in, how long they're either nursing or taking a bottle, and how many diaper changes you're having to do (a good measure, sometimes, of dehydration).
Look for signs of dehydration, such as dry cracked lips, very few tears when crying, and the color of the whites of their eyes (sometimes dehydration can cause the whites to appear a little gray-ish).
Make sure you have the doctor's name and number (the one who treated them in the local hospital) in case Children's needs to contact him for records, observations, etc. You think you're going to remember all this stuff, but it's amazing what flies out of your brain when you're being asked urgently "what was the doctor's name who discharged them because we need to see his records" and you can't believe that you can't remember. Have the local pharmacy's number too (where you got the antibiotics from).
Don't get yourself too stressed (I know, easier said than done), just keep trying to keep them rested and hydrated. Let the basic chores go (use paper plates and never mind about polishing the woodwork or extra things like that). Take care of yourself, and keep yourself hydrated and fed, and put your feet up when you can. If someone offers to come help you, let them.
If after a reasonable time they don't seem to be getting better, prepare a bag to take to the Children's Hospital. Medications (yours and theirs), comfortable shoes and clothing for you, a change of clothing or extra underwear for you, phone/tablet/laptops and their chargers, a book, wallet with insurance cards and debit cards, etc. If you end up not needing it, it's no big deal to put the stuff away when the twins have improved and things have settled down.
Keep in touch with the local hospital. Find out if that Children's Hospital has an ER and a nurse you can speak with. If you have insurance, your insurance company may have a nurse you can consult with.
I understand - my daughter was 4 and we were living in northwest Kansas, and she was really, really sick. The only hospital wasn't even a real hospital - it was a "medical center" with offices and some basic equipment. Denver was 4 hours away, Wichita was 6 hours away, and Nebraska choices were a little closer but not much bigger. We ended up having to go to Denver, and I wish I had been more prepared (baggage-wise). It was a mad scramble to remember things to pack, and there were things we forgot and wished we had brought.