Modify my old (1978) car's existing dash light dimmer (10 ohm Rheostat) to control LED replacement globes
The 10 ohm Rheostat will be in series with the light, that would work with light bulbs.

Suppose the light bulbs are 10 Ohms in total (that's 10 ohms for all of them in parallel) then with Rheostat = 0 ohm you get full power and with Rheostat = 10 ohms you would get about 1/4 of the full power. (1/4th since the voltage is halved but then the current halves as well).

The LEDs consume much less power so less current flows so less voltage is dropped. Suppose the LED us 1/10th of the original light bulb current then the voltage drop over the Rheostat is also 1/10th. So 6 V (12 V / 2 for bulbs) becomes 0.6 V. That's useless for dimming !

So if we could increase the current back to it's old value then dimming would be much better. What you could do is connect light bulbs in parallel with the LEDs. Those give light and might break. Instead of bulbs you could use resistors, these behave similarly to bulbs but give no light and should not blow either (assuming you're using them correctly).

So what value resistor should you use?

That depends on how much current was flowing originally, we do not want to exceed that current as that will damage the Rheostat.
Suppose there used to be 4 light bulbs of 2 W each that is 8 Watt in total

8 Watt / 12 V = 0.67 A then an equivalent resistor (to "fake" the load of the 4 2 W bulbs) is 12 V / 0.67 A = 18 ohms

Since the power is 8 w you need a 10 W resistor of 18 ohms.

If you do not have one of those lying around, try a 12 V 8 W (or 10 W) bulb, connect it in parallel with the LEDs and see how that works.

Note that this isn't a energy efficient solution ! Normally we use LEDs to save power. My proposal is an "easy fix", it is not intended as an energy efficient solution. Then a car isn't energy efficient anyway. Also the 8 W for dashboard illumination pales to the 2 x 50 W of your car's headlights (assuming they're still bulbs).