Take a look at how many bars you have showing, at the top of your phone screen.
1 or 2 bars is low signal and could explain why there's a delay before the text messages are sent out. Is this the case?
If you have over 3 bars, there shouldn't be such a delay. What network are you connected to when this happens? (LTE, 4G, E, etc)
If you send yourself a text message, how long does it take before it's sent/received?
Just to clarify, do you mean it can take up to 10 minutes before your phone shows your text as "Sent"?
(You can tell that your phone is attempting to send a text message for up to a full 10 minutes..)
Or does it take 10 minutes, once it's sent from your phone, for the receiver to get it?
Thanks for clarifying