I have my moms old phone and we have family plan but she is getting all my texts on her phone and mine how do I fix that

No guarantees, but the following might help.

On your phone......your mom's old phone.....

Tap Settings

Scroll down and tap Messages

Tap Send & Receive

A check mark should appear only next to your phone number

On your mom's new phone.....

Tap Settings

Scroll down and tap Messages

Tap Send & Receive

A check mark should appear only next to your mom's new phone number

Are you logged in with the same Apple ID?

Manage Family Sharing settings – Apple Support

Check in Settings>Messages>Text Message Forwarding> toggle off other devices

Sounds like you are both using the same Apple ID. Each person should have their own Apple ID.