"Seldom did my uncle get sick just before he died." is not a good translation. It means that my uncle seldom got sick in the short period just before he died. This is not what you want to say, because he probably did get sick in that period.

The correct translation is:

"Seldom did my uncle get sick until just before he died."

but this is still a very unnatural word order. The natural translation is:

"My uncle seldom got sick until just before he died."