I just want to say that I have the exact same problem and found a much simpler but slow solution.
Like OP, I also sent a bunch of messages to a bunch of groups. Every other answer gives the same solution: Reinstall WhatsApp.
I then figure something out. We both have sent multiple messages to multiple groups. So maybe WhatsApp is still trying to send those messages and I cannot send any other messages until what I send is there.
Solution? I just sleep. The next day the problem is solved. No need to reinstall WhatsApp. Just sleep. Put my HP on a WiFi connection. Just wait. WhatsApp is sending all those messages to thousands of people. Once done. It's done.
Update: The solution is not good enough. Yes, eventually it works. But now WhatsApp delays my messages again. I do not mass send messages anymore. It seems that this is either a bug or a feature.
WhatsApp Web says:
Phone not connected. Make sure your phone has an active internet connection.
Any message sent is delayed. There is a clock If I try to send messages via phone, the message is not sent. It's as if my phone is not connected to the internet. It is.