What is the percentage of single women on welfare? According to whom? What percentage of those tout themselves as independent, vs. those who recognize their need for aid? How does that percentage stack up against the number of women who are indeed independent, including the ones who aren't particularly vocal about it, and those who don't *need* a man but choose to have one?
Numbers can absolutely lie, if they are 1) taken out of context, 2) misconstrued, 3) collected/calculated improperly, or 4) downright fabricated.
Since you are one of the people on the front lines of government assistance (and you have my utmost respect for that - I'm sure it's a thankless and frustrating job), I'd suggest you to keep a personal, anonymous, unofficial tally for a month. Count the number of people who you see in your office that fit your description above (i.e. have a blatant outward show that the benefits they receive are either misused or unnecessary), the number among those whose applications are successful, and what percentage of the total benefit-receiving population your office serves is composed of that group.
I don't need you to post your findings, nor should you if you value your job. I just think it might be interesting for you to compile your own numbers, that you know you can trust, and when the month is complete, for you to take a step back and look at them.
Granted, even this tally doesn't take into account what each person is dealing with (and therefore isn't completely reliable in assessing whether they actually need benefits), doesn't track how long each person has been taking and will take benefits, and isn't representative of the nation as a whole (or even the state or Congressional district in which you reside). Still, you'd have quantifiable evidence for or against your impressions of your community.