If it's considered bad form to resurrect year-old threads, then please slap my wrist (gently, please; I'm new here!)

A fairly simple explicit example of a "sumpact" module that is not f.g. is as follows.

Let $R$ be the ring of functions from an uncountable set $X$ to, say, a field $k$. Let $M$ be the ideal of functions with countable support.

Then it's very easy to show that $M$ isn't f.g., and fairly easy to show that it is "sumpact", using no set theory beyond the fact that a countable union of countable sets is countable.