My completely vanilla copy of Crysis Warhead had my guns start shooting chickens after a while.

A quick Google search said that was something to mess with software pirates. There's just one thing: I didn't use a pirated version. It happened with the version I got from GOG.

I ran it in a 32-bit Wine prefix. Has anyone else run into this?

See here why this happens: https://www.gog.com/forum/crysis_series/notice_temporary_removal_of_crysis_warhead_32bit_version