First up, Dithering is your friend. It will certainly make gradients and highlights look better.
In your Export preview, check the different dithering options to find the optimal setting. Depending on the image, using a different dithering — "Diffusion", "Pattern", or "Noise" — can look better (or worse).
Next, make sure you export the gif using "Adaptive" with 256 colors for optimal color usage.
Last but not least… related to the fact you say that export does "distort the colors", I would like to point you to the color profile you are using. GIFs (in contrast to JPGs) do not support color profile embedding, so GIFs will always fallback on sRGB. To prevent related color-shift issues, check your Photoshop "Color Settings" (via Edit menu, or with Shift+CTRL+K) and make sure it is configured to use
sRGB IEC61996-2.1 by default in "Working Space — RGB".