I'd like to thank everyone for their help - the problem is now fixed. For the sake of others with this problem, here's what was going on:

For some reason, while the browser was receiving a Cookie (I turned on the "Warn Me" option), it wouldn't actually save it when I tried to send a Hash. I fixed that by sending 2 cookies, each with a simple scalar value.

However, the other problem was that while the CGI docs claim that if the "path" parameter is left out it gets set to "/" (allowing all your scripts access to the cookie), I found that it actually set it to the full script path (which didn't work for me since I was using 2 scripts). I had to manually set to path to "/".

These were both solutions I tried before posting, but I guess that it wasn't at the same time. And while no one actually literally suggested this solution, some of the advice prevented other bugs from cropping up. Thanks again!