SUCCESS! I destroyed a couple at first, but that was because the stamps were soaked through. I tried on some "fresh" stamps, and it worked well. I have found, though, that it needs to be an iterative process.
To summarize the process I just used...
1) Apply Bestine to envelope side, which soaks through to back side of stamp. I prefer to use a clean cotton swab, which I reuse until "dirty" with adhesive. Cover entire stamp with Bestine so you can see it through the wetted paper. Carefully, but quickly remove stamp from paper.
2) Place stamp face down on suitable surface - preferably glossy, non-hygroscopic, like the glossy stamp drying book pages.
3) Re-apply Bestine to better than half of the backside of the stamp. Gently hold down close side of stamp with finger from one hand, and proceed to gently "push" adhesive off with credit card edge (one direction, away from you) with the other. Maintain light, uniform pressure at first. Rotate stamp, and repeat for the other half. This is the first rough pass to get 90%+ of the adhesive off. Directionality is key. Work from the center of the stamp towards the perfs, and only in that one direction.
4) Wipe credit card clean with cloth, and repeat step 3 as needed to get last bits of adhesive. May need to "work it" with the cotton swap a bit to get stubborn areas to come loose. Keep the credit card and the substrate you are working on clean to avoid "re-contaminating" with adhesive. You can use the solvent to clean the credit card, too, if needed.
Now for all the US stamps I tried, this was good enough (after about 3 or 4 iterations). The tackiness was sufficiently gone that I could touch the back of the dry stamp with my dry finger, and it would not stick. There were some French Marianne self-adhesives that were more stubborn, however. The adhesive was effectively gone, but a hint of tackiness always remained. Any more cleaning, and I would damage the stamp. In this case, I opted for an ever-so-slight amount of talc. I figure that since the adhesive was 99.something% gone, the odds of adhesive creep around the talc, and future problems was minimal, but it was just what it needed to take away that last residual tackiness. A last resort, so to speak.
I don't know, what do you guys think?
I presume one could substitute the Citrus or other solvent for the Bestine.
Thanks for the tips, Fred!