If I get monthly VA service-connected disability compensation, can I still get my 2013 taxed severance pay back without it effecting my monthly VA Disability check?

Yes you can. The only thing that affects your disability pay, is your disability status - which I understand is presently 80%.

For the money you're talking about, are you referring to money paid you by DFAS which you paid taxes on, that you later learned via a letter from the VA could be treated as "tax exempt retroactive" VA pay for tax purposes?

Well, I was told during M.E.B. that if I wanted to receive a full VA disability check  that I would need to pay taxes on my severance pay. When I got out I went to a local CPA and had it taxed.

Yep, you were told right. But I don't think it would matter for state taxes. Only federal taxes. If I were you, I'd consult with the base legal office on a base closest to you, that is in your state. They have tax-pros there this time of year that would be most familiar with the state tax stuff. Heck, if there's an RAO (Retiree Activities Office) on that base or near by, my bet is they have tax pros there too. That would be even better!
But bottom line is, on the federal taxes you were told right.

Carl, even under The Combat-Injured Veterans Tax Fairness Act of 2016?

That's why I referred you. I'm not familiar enough with the Act to be of any assistance with any degree of certainty on my part. I've only got a 10% disability, so the act doesn't apply to me, and therefore I've never read it.