I have an old house with mostly k+t wiring and in one room only both duplex
outlets read "open ground" with my tester. All other outlets in the house
read correctly.The only other duplex in another room that is also on this
circuit reads correctly. I believe this circuit runs most of my overhead
lights.

When I test with the multimeter I cannot get 120V between hot and the
ground socket either but when I meter continuity between the neutral in
the central light socket and the neutrals in each duplex they show
connectivity so.....

any ideas?