Public intoxication is normally a Class C misdemeanor and should not be an issue. The DWI first offense is usually a Class B misdemeanor and will disqualify you for the 5 year period. The only gotcha might be is the PI was a class B for some reason then you would fall into the 'chemically dependent 10 year rule
Is not a chemically dependant person (a person with two convictions within the ten year period preceding the date of application for offenses (Class B or greater) involving the use of alcohol or a controlled substance is ineligible as a chemically dependent person. Other evidence of chemical dependency may also make an individual ineligible for a CHL),
So, if the PI was a C and the DWI is over 5 years and there are no other offenses, you should be OK. Hopefully you listed them and sent documentation to show the disposition?