After a few days of trying and fiddling with antennas and radios, I finally got the contact made with the guy who needed me for the last county on the list. I've still got problems; there's a horrendous noise level on 40 meters, and I don't hear very much even on higher bands where there's not as much noise. It'd be nice if it worked reasonably well.
There are a few more things to try, but the pressure is off. I'll get the guy the proof he needs of the final contact in the next few days, and he'll collect his award at the ARRL National Convention in a few weeks.