Can't transfer AirTag to new iPhone 12 (with same Apple ID as my old iphone 8

I transferred all my data from old iPhone 8 to new iPhone 12 but my two AirTags didn't transfer. I am using the same Apple ID on both phones. Using the FindMy app I've tried to install the AirTags on the new phone but the phone is just searching and searching in vain for the first Air Tag, which I've placed right next to it. (I obviously can't pull out the tag because it's long gone--my cat has been wearing this AirTag for about 6 weeks.) I shouldn't need to remove the Tags from the old phone before I can add them to the new one as long as both phones have the same Apple ID--so I've read in one of these forums. But it isn't working. (I do plan to wipe the old phone clean, but not till I'm sure the AirTags will link to it. And I don't want to buy new AirTags--I've already had to buy two new ones after both cats lost their crappy collars after only 4 weeks; they have better collars now.) Any suggestions? Thanks!