I've seen this issue a lot too, and infact had it myself. Heres what I did to clear up the issue. I'm sure not all of these steps are warranted, and in fact, I'm not sure which step in particular fixed the issue...but I recommend you do them all, just incase:
First I went into Settings > Applications > Manage Applications > Handcent:
There I cleared data, cleared default, force stopped, then uninstalled
I know you don't have to clear data/default and forcestop if your just going to uninstall; but just bare with me and do it anyway.
Next, go back to the apps list and hit your menu soft key at the bottom of your phone. Select filters, then select all. This will display system apps as well. Go to the stock Messaging app. Clear data and force stop it.
Now go to the market and download handcent again.
Go to your stock messaging app and check that all of your settings and notifications are configured to your liking.