Aw...toes are so much fun to deal with
We had a recurring injury keep popping up in one of our greyhounds toes. He had surgery last Friday to remove the dead tissue and stitch back together the healthy tissue. We did one round of antibiotics seemed to clear it up, but then it came back several weeks later. Sometimes the dead tissue deep in the toe, debris or just deep infection can keep it from healing properly. Since dogs use their feet/toes all the time, every day that tissue constantly has pressure on it so it can be hard to heal.
One of my other greys may need two toes amputated due to severe dislocations and some arthritis that's developing. We are really trying to avoid that. If it were one toe, no problem - go ahead and lop it off. But two toes in a young, still very active dog I'm much more hesitant to be hasty with amputation. She goes back to the orthopedic specialist next week for a reassessment after two (very loooooong) weeks of crate rest and leash walks.
We adopted a dog once and she was missing a back toe...never realized it until I went to trim her nails and came up one nail short! Dogs can live just fine with a toe missing.
I would want an xray done - If it's purely an issue with the soft tissue you probably aren't going to see much, if anything. But, they're relatively inexpensive ($60-$75 around here) and will give you a more complete picture of what's going on with the bones inside the foot.
If you've now done three rounds of antibiotics and that hasn't resolved it, it's very possible there's now infection in the bone. At the very least you have some seriously infected tissue there. I would most likely opt for surgery to remove the toe without too much hesitation. Bone infection is not something to mess around with.
However, if you do not feel comfortable with your vet - get a second opinion. They may be able to refer you to a orthopedic specialist...I really prefer to have my dog seen by a specialist with toe/foot issues if it's anything more than a soft tissue injury.
Good luck...keep us posted!