Had the same problem and just telephoned 119 got through to a person in less than 5 mins and they found me a date that had a second jab available. The message online told you not to phone them but I physically couldn’t continue to book a first appointment without the system finding a second one. It was lose your appointment and start again the only option. Tried it a third time and it offered me Southampton that’s when I gave up and phoned had it all sorted in 10 minutes.