Can I get texts messages forwarded to a different phone

You can forward texts individually but you can’t setup forwarding of all texts through ATT but if you’ve got an iPhone, you can set up text message forwarding on the iPhone. 

Don’t think that’s doable 

No. There is no mechanism to forward text messages. Only phone calls can be forwarded.

If you have lost or damaged a phone and have not yet replaced it, the SIM card is how you receive your calls and text messages. You can move your SIM card to a different phone.

⚠️Just be aware that if you are making an insurance claim you must make your claim before you move your sim card.  Asurion covers the phone on the line at the time you file the claim.