How do I safely use an extension cord in wet grass?

Yes, do both. GFCI is the non-negotiable safety net. Then physically keep the plug junction off the wet grass by putting the connection inside a small plastic container flipped upside down with the cord coming out through a notch, weighed down with a brick. The real risk is where two plugs meet—water can wick in and cause a short, so keep that joint dry and elevated.