Welcome to iMore!
A few things to check:
Settings -> Sounds -- make sure a tone is selected. Maybe change it and see.
Do Not Disturb -- are all text alerts not coming through or just for certain texts? I use DND on a group text (shows a moon next to the text string in my list of conversations). You can turn it off by going into the text thread, tap Details at the top right, and scroll down to Do Not Disturb.
There may be other options but those are other items to check beyond what you have done already.