I just visited the Midway in San Diego this spring. The ship has been turned into a museum, and the Bird Dog is one of the first exhibits you can see. It’s an amazing vessel, with lots of awesome and terrible stories to tell. Veteran crew are always aboard to answer questions about the ship and their service. HNers might especially like the CIC and other information handling rooms — they’re configured like they were during Operation Desert Storm.
The Midway is definitely the best in the country, but if the west coast is too far of a trek there are similar museums within decommissioned aircraft carriers in Mobile, AL (USS Alabama) and Charleston, SC (USS Yorktown) that I'd recommend. Each has interesting histories and exhibits, and veterans in attendance.
Also Intrepid in NYC, or do you not recommend that for some reason? I enjoyed when I went, but that was over 20 years ago now. They've had a major restoration and put one of the Concordes on display since then.