No "Text Message Forwarding" option in iPhone Settings
Finally figured this out, so posting in case it helps someone else.
Background: My Mac and iPhone had messages synced. I got a new phone (same number), and suddenly my Mac could no longer send/receive texts. (iMessages to other Apple users were fine, but not regular texts, e.g. to Android phones.)
This Apple article had some tips. (How to forward SMS/MMS text messages from your iPhone to your iPad, iPod touch or Mac - Apple Support)
The tricky part was where it says, "Don't see the Text Message Forwarding setting on your iPhone? Go to Settings > Messages, turn off iMessage, turn it back on, tap Send & Receive, tap Use Your Apple ID for iMessage, then sign in with the same Apple ID used on your other devices."
I tried following that, but when I tapped "Send & Receive," there was no "Use Your Apple ID" prompt. It fact, it already showed my Apple ID logged in. In very tiny blue highlighted font at the bottom. Tap that, sign out, then sign back in. That worked for me right away.
Hope it helps!