SMS sending to some contacts but not others in Messaging app

I have a Nexus 5 with Cyanogenmod 13.0.

I use the built-in "Messaging" app for SMS. I've never had a problem with it until now. I have been able to send texts to any number I've tried.

A week or two ago I switched phone network. Since I did that, I gradually noticed that though I can receive SMS from any number (as far as I can tell), the messages I send only get through to particular numbers.

There's no pattern that I can find with the numbers which work and which don't work. They're all numbers I've been OK with in the past, and both working and not-working numbers seem to exist on a variety of networks.

The message briefly says "sending" or something, and then soon after that they show as being sent, and I can read information about the message (like the timestamp when it was sent, and the number it was sent to, and all of this looks legit).

I phoned my new provider and after some time they told me that there was no record of their end of the disappearing messages ever getting to their network. They said it must be my phone.

I didn't want to believe this, since sending to these numbers was fine on my old network, but tried my sim card in a friend's phone anyway. Lo and behold, again using the built-in Messaging app, the texts sent just fine. And using his sim card in my phone, I could not send to that number. His sim in his own phone sent OK. So it did look like my phone after all.

This doesn't make sense to me, and I started looking for possible reasons. I found this answer to another question which suggests trying a different SMS app.

I tried Google Hangouts and sure enough, it can send the message to this number. But I don't want to use Google Hangouts for SMS, or any of the other alternatives.

What can I do? Is there anything I can try which may help, without wiping out my SMS history, or reflashing the rom, or switching rom? I just find it so bizarre that suddenly I can't send to these particular contacts, while nothing I'm aware of has changed other than my network, yet it doesn't seem to be their fault.