the underseat area would not even allow us to stow a 12"x12"x4" fully packed backpack.
That might be why Delta does not specify maximum dimensions for personal items.
A month ago, I flew Delta domestically (Denver to Seattle). Because it was the Thursday before Memorial Day, I thought the flight might be crowded, and I would have to gate check my carry-on bag. Therefore, I used my "old" Goodhope Travelwell bag (18"x12"x6½") as my personal item for essentials (one change of cloths for church on Sunday, sink washable underwear, toilet kits, and netbook), things I absolutely had to have in Seattle, and I left everything else in my normal sub-regulation carry-on.
I had no trouble putting my personal item under the seat in front of me (and there was room for my carry-on bag in the overhead bin, so i didn't have to gate check it). I didn't stuff my personal item, so it was probably less than 6½ inches thick, but I'm sure it was more than 4 inches thick.