I'm not exactly sure what I did to do this, but now everything seems to be working. I noticed the change after I added a second A record for mail.bladescape.com that points to my internal LAN IP address (the first record points to my external router IP address).
After I did this I could send and receive e-mail with no problems. Again, I have no idea if this was the only factor that caused Zimbra to start working properly, but I'm certainly pleased that it's now working.
I'd like to thank everyone who made posts on this topic for their suggestions, advice, and assistance in helping me solve this problem. I appreciate everyone who posted to this topic and am sure that more than one person was helpful.