Well, it looks like the automatic upgrade problem was not WordPress, but my hosting company 1and1.com.
I searched around a bit more and came up with this excellent post by Michael Stults which fixed the issue. Thanks a bunch Michael.
I won’t rehash all that Michael has gone over, but it seems that 1and1.com was serving up PHP 4.0 instead of PHP 5.0, which WordPress was looking for.
A quick paste of these two lines before the closing tag of the .htaccess file makes it use PHP 5.0 and problem solved.
AddHandler x-mapp-php5 .php
AddType x-mapp-php5 .php
That’s all it took. Now all four of our WordPress sites are working right and automatically updating as they should.