Your instincts are right. When a turtle can't submerge, or floats "crooked" it has either a respiratory infection OR pnuemonia. You will want to take it to the vet to be xrayed as soon as possible.
The vet will probably administer antibiotics via a needle. Have the vet show you how to inject the turtle as many times as it takes until you are comfortable doing it yourself, or leave the turtle at the vets until it is well. If you live close enough, you can run the turtle in for its shots.
Clean the tank out really well, and make sure to change the water in it as much as you possible can.
Good luck with this,
PHKiwi