I've used https://play.google.com/store/apps/details?id=com.riteshsahu.SMSBackupRestore&hl=en successfully in the past.

Edit: Beaten to it. Great minds.

I've always found using the built in wizard on the new device reasonably good now...

Huawei / Oppo use PhoneClone

Samsung have their own. 

Not often they don't put you through a wizard to migrate all that sort of stuff.

They just do a mass BT/local wifi xfer

What brand is the new phone? Both OPPO and Huawei have a "Phone Clone" app and you can easily select just SMS and have that copied over.

Backup to your Google account and just do a restore.... or maybe it was the Samsung account that they did that - one of them did.

I did the default Android clone option when setting up the new phone. Most things came across , not txt's  . Maybee because the new ph didnt have a sim when I did this ?
LG to Moto7

I'll try the app recommended above

I did find a way to extract old versions of apps off the old ph & install on the new ph , via Airdroid .
Some old app versions are better the newest versions (and some apps no longer on the play store so didnt come across on the clone )

Cheers