Epilogue:

I spent some time yesterday looking at gas dryers, and I realized it'd actually be closer to $500 including delivery to get a gas dryer with the same features of my current electric model, the most important of which being a moisture sensor. Staring at a $500 investment for little-to-no gain or $1000 to replace the panel, I decided to give one more electrician a call.

He came out and looked at my panel and told me that he could add the dryer plug to one of the existing upstairs circuits that didn't have a lot of load on it. Things seem to be running smoothly now, and I've got my dryer going as I type. He did say that this is the end of the road for this panel, both in terms of load and in terms of the physical shape of the panel.

He did the work for about 1/3 of what the dryer would have cost me, so if I squeeze some years out of the dryer, it was a good investment, and if I never have to replace my breaker panel, it was definitely the right choice. We'll see, but as someone who just shelled out big paper for down payment and closing costs, I think this was a decent stop-gap measure. Thanks for all the advice and info, everyone.

Now, to see if the Christmas lights push this whole darn thing over the edge...