But, before using any silver polish or tarnish remover, try the Baking Soda and Aluminum Foil process as documented by Brian DiPalma.
The silver plating on these old razors was thin to begin with. I have had good results from this process, which converts silver tarnish back into silver metal and preserves the plating. Polishes and tarnish removers physically remove the tarnished plating, sometimes down to the base brass metal underneath the plating!
I have the Hagerty's silver dip. It works well, but it doesn't cut through the filth on pitch black razors. On silver razors, I usually do the hot water/dish detergent soak, then a little polishing with Blue Magic, THEN the Hagerty's dip, then a thorough cold water rinse, and then Scrubbing Bubbles to finish.
What Hagerty's is great for is maintenance. Once you've gotten your silver razors gleaming again, a periodic 8-10 second dip in the Hagerty's will keep them looking as clean as the day you restored them.
Oh, also, just for a little peace of mind. Hagerty's silver dip was recommended to me by a 25-year metalsmith who knows everything there is to know about metal. She said it is completely safe and will not remove metal at all. Blue Magic is also non-abrasive, I believe. And I can also vouch for the baking soda and aluminum foil method mentioned above. It's almost magical in its effectiveness.
Have yet to try the baking soda and aluminum foil method, but probably will next time I rehab a suitable razor as it has the benefit of removing the sulfur without loss of silver.
Nonetheless, I suspect the amount of silver tied up in the tarnish is minuscule, even by the standards of the stingy plating used on most Gillettes. I've used Hagerty's polish (not dip) on flutes for many years without scratches or other harm. I used it on two extremely tarnished (black) razors, and it was a long slog but very effective in the end. Plus another that was mildly tarnished (very quick and easy).
Hagerty's doesn't appear to be abrasive. It does its work by chemical reaction. The current version claims to leave a tarnish-inhibiting residue, so it may be good as a final step even if you do use sodium bicarbonate and foil.
Here's an added tip: You can buy tarnish-preventing paper that needs to be replaced every 6 months. If you cut a handy bit off and put it in a fairly airtight container with the razor, that seems to help. Works like a sacrificial anode I suspect - binds up the sulfur. Just keeping it sealed up and out of urban air with the prevalent sulfur dioxide will help too. And don't lay silver down on newspapers - the cheap pulp contains lots of sulfur compounds...can't prove it'll harm anything but I suspect it will and have avoided putting silver flutes anywhere near the stuff. Good luck!