If I have the shoulder DUIs on my dress blues, would it be appropriate to wear the Stetson or just keep the regular barracks ASU cap?

The stetson is not an issued item and isn't covered under 'official' uniform regulations. As an unregulated "traditional" uniform item, the wear of it is prescribed and proscribed by the unit commander.

Probably the "highest level" of "official guidance" comes from the 1st Cavalry Division* Commanding General but keep in mind this only applies to 1CD - however, since they are the 800lb gorilla when it comes to tradition, many follow suit in what they do (plus they are the ones that brought back the traditional wear during and after the Vietnam War*).

As the 1st Cavalry Division Association puts it - "All former Cav Troopers may wear their “Cav Hat” at their own pleasure but are encouraged to wear them to all gatherings of Cavalry Veterans and unit reunions. The “Cav Hat” is always appropriate at patriotic observances like Veterans Day, Independence Day and Memorial Day. Of course, never wear your “Cav Hat” in circumstances that would bring discredit to the Cavalry or your unit."

You'll get answers along the lines of "You're not in the military, do what you want" or "Once Cav, ALWAYS Cav", but I applaud your trying to figure if it's wear by a veteran is in line with the tradition of the "Hat and Spurs".

Would you be keeping with tradition? The 800lb gorilla thinks so.
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* 1CD MOI on wear of the stetson and spurs - https://1cda.org/wp-content/uploads/MOI_August-2021.pdf
* "Cav Hats" - the story of the Stetson - https://1cda.org/wp-content/uploads/Bill-Gillette-Story.pdf

I have served in two Cav units. I always wear my Cav Hat, black boots (highly polished black ropers), and combat (gold) spurs any time I'm in mess or dress blues. (Even my saber at weddings.) I have never been called out for it. Nor have I ever been in a formation wear it would not have been appropriate. I like COL Cudworth's comment and guidance; "At their own pleasure" covers it.

You're retired, do what you like with your Stetson...