My new iPhone 15 won't let me download app without my husband's app account.

If it needs your husbands apple account it is most likely the case that your phone is linked to your husbands Apple ID. Apple ID's are strictly personal,.so you might want to consider to create your own Apple ID and to logout of your husbands Apple ID on your iPhone

The apps were likely downloaded using your husband's account, or possibly as a benefit of being linked to your husbands account if you had set up Family Sharing but now no longer use it. You need to either delete the apps and redownload them with your own account, or set up Family Sharing so you can download them (not possible with all apps) as being part of a Family Sharing group. To learn how to set up Family Sharing, read this support article --> Set up Family Sharing - Apple Support