I have been having exactly the same problem. I stopped receiving text notifications of booking requests from Airbnb in November. This is a real hassle for me, as I live in an area where my internet is sketchy, also I work during the day, so can't be constantly checking the Airbnb site. When i received the text notifications, it alerted me to get online so I could respond within 24 hrs.
I reported this to Airbnb over a month ago. Have an "open support case" going and have exchanged about a dozen messages with the support person, who tells me this is an airbnb glitch and that the technical team is working on it, but as of now, I am still not receiving text notifications- it has been 2 months now that this hasn't been working and I am beyond frustrated. How hard can this be for airbnb to fix? I feel they are actually doing nothing about it, just telling me they are.
All my settings are as they should be, and nothing has changed on my end- same phone provider, same phone number, etc.
Sorry I can't help, but at least you have some moral support....
Also, start a dialogue with airbnb support so they get it that this is happening to other hosts.