What should I do in these situations?
Unfortunately, all you can do is keep trying...just like when you try to check on via voice and don't get an answer. After a few attempts, go back to the previous frequency and let them know you couldn't make contact. Or, if you were coming from UNICOM, try to reach any OTHER controller you can and advise them "I was supposed to reach 127.97 but the controller hasn't acknowledged after a few attempts, confirm you're able to see this message?" or similar. I suppose you could also try a private message to the controller, though chances are if they're not watching their radio message, a PM might not get much further.
If you're flying as "/r", which is a perfectly valid thing to do, then the controllers are responsible for monitoring their messages and responding to you. Of course, ensure you're sending an actual radio message and not a private message when you do check on. (Sounds like you were doing it right.)
As others have said, it can get very busy and sometimes the radio message doesn't get our attention as much as it should. It's not a great situation but it's also understandable when you're handling the volume of traffic we see during FNOs; it often exceeds what a real-world controller would be handling.