Afraid she's right. The spot on drops go on behind the neck of the dog and last for about a month. You can get injections which last a lot longer but they are therefore more expensive and only available from vets.
It is possible to do without, but you'd need to groom and check daily for fleas, and possibly limit where the dog can go when it's out (other dogs, long grass, etc) You would still need to treat in the fleas most active season (summer) and in any case, prevention is better than cure, as once fleas are in your house, carpets and bedding, they are notoriously difficult to get rid of.
Neosporin works for people and dogs, but dogs will lick bandages right off, and sometimes eat the bandage. I think Neosporin when ingested also gives the dog the runs. It just happened to my dog who is recovering from surgery.
The best thing is really an Elizabethan Collar. He will hate it, but that's almost the only way to stop him from licking. Well, you could sit there and stare at him until he heals, but I assume you will need to sleep.
I hope you gave the groomer a swift kick in the shins.