Today, I powered up the third laptop, let it run for a while, and then went to the AvastUI. I entered the my.avast.com interface, and then did exactly what I did the day before; click on "Use license". This time it worked and now shows expiration of 10/15/2018 as it should. Chalk one up to persistence. All laptops now have the correct date when checked on my.avast.com, also.
I'll wait until the old license expires and then request a copy of License.avastlic for the current new subscription. I like to keep it on hand in case I have to do a clean Avast install. It should have the correct expiration date, one would think.
Thanks very much, Moroni. Also thanks to Redman and Eddie for you input.