Quote: " DWFS will only use that as verification with my employment with that company for some unknown reason."
Okay, I have spent many years of my life verifying incomes for state and federal income based programs. And NOTHING in these programs assumes, if your past employer refuses to verify a quit date, that you are still working. There is not only one way to verify that you have quit a job. The legal action you could take is to appeal your denial, when you received it.
Through the DWFS, you appeal the decision to deny your eligibility. And there are other means to go through to get your verification of the last day you worked at the previous employer. If not, there's no way that anybody would ever be approved for benefits, because many past employers will not cooperate with this one specific system. Some employers won't cooperate with anybody. They have to provide other ways to be certified.
I am positive that your previous employer was submitting their employer taxes and quarterly reports to the unemployment system, they are verifying who does and who does not work for them on several other state venues, and you cannot immediately jump to "suing them" for keeping you from qualifying for a state program, even if you had the money to hire a private attorney to do this. Speak with legal services in your area about filing an appeal of your denial of benefits, if this is what has happened