Lots of accusations, no evidence. HSUS has many critics and some appear fair but I don't see anything in their history that was campaigning.
What 501c3 organizations can do is define issues. Both liberal and conservatives churches do it all the time. Some cross the line, others balance very well.
This is not a free speech issue because no one's speech is being taken away, even if straddled because you get the non-profit tax-exempt status you choose to get. It isn't like you can't do that work without that status.
Organizations like Media Matters which is a highly partisan (for the left and not necessarily for Democrats) who have a mission to point out right wing misinformation. What they do is not considered charitable in the strictest sense of the word but does not violate IRS rules which is a position held by one of the first people who challenged their status. They dance close to the line but have smart lawyers who know where the line is. (The are several groups from other perspectives that do similar things).
The biggest problem is that law is not black and white on purpose.