With a bit of luck, they do. Especially as you have seen a vet promptly.
Please be aware that any antibiotic is not an instant fix; unlike painkillers they build up over several days until they reach optimal efficiency and then wear off gradually. Optimal efficiency it reached towards the end of the course although you may start to see first improvements within a day or two. That is the reason why you should always finish a course of antibiotics and not stop as soon as symptoms disappear.
Please make sure that you give probiotics either one 1 hour before or two hours after the antibiotic as doxy is one of the antibiotics that can impact more on the gut microbiom so you want to support it.
Probiotics, Recovery Foods And Vitamin C: Overview With Product Links
Please don't panic over the distance. My own specialist vet is an hour away; with a traffic snarl on the main South/North motorway/highway, it is easily 1.5 hours or more. I have travelled longer on the train with freshly operated or newly adopted piggies. They have all survived just fine.
Here are our practical travel and vet visit tips:
Tips For Vet Visits
Travelling with guinea pigs