Can I use an extension cord in wet conditions without GFCI?

You shouldn’t. There is no safe way to run a regular extension cord in wet conditions without ground-fault protection. Even if the cord is rated for outdoor use, the risk of a ground fault is too high. If you absolutely must, use a heavy-duty, three-prong cord rated for wet locations, keep connections off the ground, and plug into a GFCI adapter—but honestly, just buy a GFCI adapter or a battery-powered tool.