I would expect the post-nasal drip is connected in some way. Given everything you describe it does sound like some kind of bacterial activity in the mouth. Of course we all do have mouth bacteria normally but I imagine that just like with gut bacteria it can get out of balance and the "wrong" types can overgrow the others. In your case it might be bacteria from the nose coming down or maybe it's some kind of bacterial infection that's causing the nose and the mouth issues at the same time. Maybe the throat infection wasn't entirely cleared by the antibiotics (symptoms subsided towards the end, but it usually takes a few more days of antibiotics past when symptoms are over to fully clear an infection).
I know you said you have dentures, so I expect chewing gum is out, but if you want to try it, sugar-free chewing gum with xylitol is known to actually be anti-bacterial. Another thing you might try is brushing the rest of your mouth with regular toothpaste at night (like the roof of your mouth, your tongue, etc.) above and beyond what you use to clean your dentures, see if that helps at all. Mouthwash might also be a possibility but many people find that to be harsh and irritating or even damaging to the soft tissues in the mouth so don't overdo it! And make sure you're getting more water than you feel like you need, often dry mouth can contribute to irritation and bad tastes.
But I suspect the source of the taste isn't in your mouth at all but in your nose and throat. Any chance you can get referred to an ORL (nose and throat specialist)?