OK, I just got through a series of email with Textender customer support, which responded very quickly and was really helpful in resolving the problem.
First, the reason why the messages show in your outgoing email box is so that they can be saved as one message instead of being broken into 140 or 160 character segments.
Second, the reason why the app wasn't working for me was because I didn't include the area code of the recipient's number. I send texts locally and when using the native app, there's no need for me to include an area code, so I didn't even consider this as an issue when using Textender. That said, when I edited my contacts to include their area code & then used Textender, all was well.
Looks like I'll be keeping this one & kudos to their customer support, which I have to say is excellent!