Try cutting off new palm leaves that have brown spots on them.
If regrowth also have brown spots cut them off too.

Because you moved into a new home, you might get better assistance by
contacting your local gardening centre.
--

Chris

Thanks Chris, maybe I should find out. We only moved 4 blocks, and from
where we moved, my former neighbour's palms are looking just spectacular. By
the same token, he's also got 3 very large gum trees providing shade for
everything below, but I'm wondering if it's some sort of bug. I'll take up
your advice on contacting someone locally!

Casey