WhatsApp performs automatic backups to your phone's storage (unless you tell it not to), with an option to back up to Google Drive with a frequency you can set. So a key question is what were your backup settings on your old device?
Your WhatsApp data should be in a folder in the internal storage of your old phone. Look for a folder in the internal storage called WhatsApp, or else for Android/media/com/whatsapp/WhatsApp. Everything will be in there: Backups, Databases and Media are the most important subfolders, but I'd copy everything to somewhere safe. Then you can try copying those things to the same locations on your new phone (but remember that that will overwrite what is on the phone currently - feel free to back that up first!). My guess is that if you replace those folders on your new phone with those from your old one you will be back where you were (minus any new messages in the meanwhile), but there are probably guides on the web that will give more detail on the procedure (it's the sort of thing I'd work out in detail if I had to do it, but haven't had to so far. It's also why I'd made sure I had backups of everything first).
No idea why your old phone can't connect though. I've swapped between old and new devices before and reactivated WhatsApp on old phones. Does your old phone connect to the internet at all?