You'd think a dimmer would just lower the voltage. Nope. That'd be a variac, and would make the dimmer weigh about 20 pounds. Actually, old dimmers use one of a couple of "hacks" to dim cheaply, at the cost of doing really weird things to the AC power waveform. These methods are extremely optimized for incandescent bulbs only. (This happens at the cost of power factor and adding a lot of line noise to your house's electrical system. if you were to hook a 120V-rated speaker to your power line, you wouldn't hear the perfect sinewave hum, you'd hear a crunchy bunch of noise, and not the pretty MBV kind. Your other equipment doesn't really like this.)
However, the LED screw-in-bulb manufacturers are perfectly aware of these obsolete dimming techniques. It is possible, though not exactly "free", to build an LED that can
- "read the tea leaves" of the weird waveform to determine what brightness the dimmer is trying to make
- include a circuit to actively allow the current leakage that old "neutral-less" dimmers require
This ability is not free. The cheapie manufacturers hedge for price, hard, and big-box sellers like Home Depot and Walmart really pin their providers to the wall to shave every fraction of a penny. So you bet those features are not included in run-of-the-mill bargain price bulbs as many consumers prefer.
Use quality bulbs engineered for this use
On the other hand, upscale vendors know there's a need and have labored to fill it. Again not a thing you'll find at the Dollar General. But if you deal with a Tier 1 supplier like GE (who has built a lightbulb or two in their history), Sylvania, Philips, Cree... so basically the Phoebus Cartel plus Cree... you will find plenty of LEDs that will do the job.
You may find them from Tier 2 vendors also, one of my faves being Ikea, whose quality is unearthly good for the price point. I don't count LoA, Utilitech and Feit, they fall in the last category IME and I have the dead bulbs to prove it.
Remember the bulbs you need should not be as cheap as the bottom shelf grab-candy. If they are, they're cutting too many corners. Playing nice with old dimmers is a hard problem and the builders need their price to make a quality product.
As far as those Tier Fell off a truck in Shenzhen vendors like you find in the old Dollar General, stop buying from them. Those products are specifically designed to sell to suckers.