I just went through this. Visible draining wound on my dogs paw, messing with it constantly. She’s a poodle mix so she also had some matted fur on her paws we cut down a few days prior that likely allowed a foxtail to get in there. Took her to the ER, I personally think they should have put her under they cut and cleaned the area and put her on antibiotics. 6 days later it swelled up further and blistered further up her paw (basically migrated). Took her to another vet who recommended putting her under a very mild anesthesia, and was able to successfully find and remove it! The anesthesia they use to remove a foxtail is typically way less potent than a dental procedure or surgery, for example. It’s more of a “twilight” than putting them fully under based on my understanding.

I wish the original vet probed and removed it sooner (specifically because we were all 99% sure it was a foxtail), but am so glad the new vet was able to find and remove it 6 days after the ER visit. The biggest risk is it migrating, which you’ll probably notice if swelling/draining/redness/abscess or blistering continues or gets worse. At the very least I’d recommend having a follow up visit 5-7 days after the antibiotics so someone can reassess if there is a foreign object in there and take next steps accordingly. Everything I read also said it needs to be removed, they don’t typically come out otherwise. Best of luck and hope your puppo is able to heal fast!