Color Shift Dimming LEDs

Dim an incandescent, and it produces a mellow, warm orange sunset glow.
Dim an LED, with the same SCR dimmer, and it flickers at 60 hertz, still producing harsh blueish light.

How can LEDs more closely mimic incandescents?

They can't. If your trying to dim an LED bulb you'll need to buy one that is dimmable which will cost more. But dimming an LED isn't going to change the overall color tempurature of the light, just lower the intensity.

If you want an LED light to mimic an Incandescent you'll need to design it to do so. Meaning it will need multiple settings. So if you want dim orageish light then the light needs to turn off the high brightness LEDs and turn on a set of dimmer LEDs at the color tempurature you want. LEDs produce one color no matter how bright they shine (to a point anyway). Best bet may be a programmable RGB LED light bulb so you can set it to the light level you want.

You can get LED's that work with standar dimmers but they cost a bit more.
The standard ones use a vary basic circuit to get the mains power down to acceptable levels, usually just a capacitor, resistor and a diode (or rectifier).
The dimmable ones use a switch mode power supply that is controlled by the PWM signals the dimmer produces.
Most also work with the cut off type dimmers used for incandescent lamps.
Only way out is to update the lamps.
For multi lamp systems you can get dimmable controllers that allow you to connect what you want as long as you stay below the max power rating.