sorry but you are screwed...
The ONLY way out is to get the case re-opened and the disposition changed to not responsible/not guilty
Even if you go back, change the plea, and get it CWOF'd, under the law you are still a PP
The chances of getting a pardon are near zero, pardons with 2A right restored are unheard of.
Can you break the law and risk getting caught with a firearm as a PP .... sure you can, there are ways to do it, but neither I or anyone here will tell you how to do it, you would pretty much be limited to owning a gun in your own home for self defense and if you ever had to use it you would be FUBAR'd beyond belief.
Did you have a lawyer?
If so he/she let you plead guilty to a first offense DWI?
That is unheard of, NOBODY pleads guilty to a first time DWI unless there were extenuating circumstances like you got them to agree to drop beastiality charges in return for the guilty plea on the DWI
a CWOF carries the exact same penalties, including the gun ramifications, but it is not considered as bad as a straight out conviction...
IF you had legal representation, and plead guilty, there is a billion to one chance to ask for the case to be reopened , the verdict set aside, and a new trial ordered based on your having ineffective counsel/legal representation, and then another one in a million chance of you being found not responsible/not guilty
in case you are wondering those odds are a 1 with 15 zeroes after it.
IANAL, the above is just an opinion, and nothing is to be construed as advocating for you to do something illegal that would get you jammed up more than you are now.