Drop these dxgi.dll files into your Bin32 and Bin64 dirs (that is, they should be beside Crysis.exe). GOG doesn't seem to want me to post any links, so please excuse the missing protocol. Just copy+paste it, your browser won't care.

ab1.nyc3.digitaloceanspaces.com/crysis_dx10_refreshratefix_v01.zip

Start the game as normal and it should run in your monitor's highest supported refresh rate. Tested only on my own machine (with a 4K60hz monitor which wanted to run at 30hz); please let me know whether it works for you.

Probably works with Warhead, but I haven't tested it.

This fix did not work for me on Crysis Warhead.

Tested on Crysis Warhead 64-bit, retail. Placed dxgi.dll into the Bin64 folder. Got 1920x1080 @24hz.

Will continue using a 1920x1078 custom resolution workaround.

Try my fix: https://drive.google.com/file/d/18gZdHhOOFHN-DTC6zk4G5U3llG52Yd03/view?usp=sharing