Obviously answers to the abovementioned questions will aid us and speed along any advice or help.
My first inclination was an issue I haven't seen too often lately but it would also depend on which device and OS we're talking about but I used to see similar issues when there were thousands of undeleted texts. This seemed to either make the app work too hard, as it was trying to make the texts available for viewing or was taking up so much RAM that the app just didn't have enough ummf to work. Either way, deleting old and unnecessary texts, restarting the device regularly (this just closes things providing a fresh restart and clears most caches) and clearing the cache and data for the messaging app usually got users back up and running. Having all of that junk more or less saved in the app caused, delayed or even unsent messages, inability to send or some times receive MMS and app crashes.
Mentioning this reminded me of one other thing, and this one I really haven't seen in awhile, Facebook and Instagram used to contribute to this issue. Clearing the cache on those two apps as well as the SMS app was important.