You've kind of answered your own question here - if you have two A records and one points to the old site, the half the time you go to the site you'll get the old one, and half the time the new one. Each DNS request will randomly return one or the other IP address. When you remove the old one then it'll work correctly!

(the browser will cache the IP address it gets from the DNS request for a while, which is why it can appear to work OK in one browser for a while, but if you come back to it later it might not!)

And for the Lions Club site, try setting the Membership Policy to Private, then the button to join the site will disappear.

Thanks

Joe

I had to delete the 'Weebly Template' from their DNS settings page and all seems fine now. Thanks.