I tried the copy to computer and then to the new mobile method, and it didn't work...
I had tried using the Google Drive method to make a backup, not selecting the videos as they take the most space, and Whatsapp first made a backup of around 400 MB, and then started uploading it to Google Drive, but got stuck at 40%. It didn't move past that for a long time, so I cancelled it. I don't know if the partial backup exists on Google Drive, or whether it can be used to restore the backup; will have to check.
I copied the Whatsapp folder from the old phone to the computer, didn't uninstall Whatsapp from the old phone, and did a factory reset. After installing Whatsapp on the new phone, I didn't open the app (as recommended in a guide). I copied the Whatsapp folder from the computer to the phone, then opened Whatsapp, and went through the process of setting up the account. It displayed all my contacts/groups, and at first I thought it had successfully restored the backup, but when I opened a group, I saw that it contained only new messages from today. It starts with xx created this group on this date, you were added, and then today's messages. None of the previous messages.
I would still like to restore the old conversations. Does anyone have any suggestions? What happens if I uninstall Whatsapp from the new phone - will it delete my Whatsapp account and remove me from all the groups, or just remove the app from the phone? What if I again copy the Databases folder from the computer to the new phone? This is if it allows me to overwrite that folder. What if I try to restore the partial backup from Google Drive? I'm going to try all these methods, but I thought I'd ask first; maybe someone has used one of these three methods to successfully correct a messed up restore job.
As
@billubakra has said, the best method is using Google Drive.