As I lean more and more from Conservative to libertarian in my old age, I find your comment most perplexing. In full disclosure, I when I disagree with librarians I usually find their economics is wrong, not there libatarian values.
A pure libertarian would say that the problem with single moms being a burden on society is that government offers assistance it shouldn't. So the problem is not Single motherhood, it is socialism. If you want to keep your illegitimate baby out of a twisted sense of self worth- fine, but don't expect others to pay. I bet the illegitimate rate would drop pretty fast.
There is a difference between a libertarian and a liberaltarian.
A liberaltarian wants a life of immoral decadence and is generally opposed to traditional values. But they also want tax cuts.