Text Message sent shows my husband’s information

Because the only phone apparently affected by this issue is your mom's, it is highly likely that your contact information (your personal phone number) is mis-labeled in your mom's phone Contacts.

If both your phone number and your husband's phone number are listed on your husband's Contact card on your mother's phone, when a text is received from either your phone number or his, it would appear to be coming from him.

If this is the issue, to correct it, delete your phone number from his contact information on your mother's phone, and then add or update your own contact information on her phone.

As a side note, iOS 17.4.x provides the option of enabling a personalized Contact Photo & Poster that can be displayed when you call other iPhone users.