I read this as: counties with US Nursing Homes & Skilled Nursing Facilities but no US Hospitals, in OH. Using US Hospitals; add "including independents" or "from Overture" for the wider Overture set.
The same question as a structured link: https://locationlists.com/find?areas=county&dataset=nursing-home-list&state=OH&b.dataset=hospital-list&has=a&lacks=b&in_state=OH

LocationLists: places per county, areas with a and none of b in OH

Set a: US Nursing Homes & Skilled Nursing Facilities, state = OH. 922 places, 922 placed in a county, 0 could not be placed. In 88 counties in OH.
Set b: US Hospitals. 6,774 places, 6,505 placed in a county, 269 could not be placed. In 77 counties in OH.

11 matching areas:
- Brown County, OH (39015): a 4, b 0
- Lawrence County, OH (39087): a 4, b 0
- Putnam County, OH (39137): a 4, b 0
- Carroll County, OH (39019): a 3, b 0
- Perry County, OH (39127): a 3, b 0
- Preble County, OH (39135): a 3, b 0
- Meigs County, OH (39105): a 2, b 0
- Monroe County, OH (39111): a 2, b 0
- Morgan County, OH (39115): a 2, b 0
- Noble County, OH (39121): a 1, b 0
- Vinton County, OH (39163): a 1, b 0

Now get the rows (counts above are free; rows are paid):
- a — US Nursing Homes & Skilled Nursing Facilities, state = OH in 11 counties: 29 places, $0.52 by card. https://locationlists.com/find?dataset=nursing-home-list&state=OH&areas_in=county:39015,39087,39137,39019,39127,39135,39105,39111,39115,39121,39163
- b — US Hospitals in 77 counties that have a: 249 places, $3.27 by card. https://locationlists.com/find?dataset=hospital-list&areas_in=county:39035,39049,39061,39113,39095,39153,39017,39093,39099,39151,39003,39007,39023,39057,39139,39039,39041,39045,39085,39089,39141,39147,39165,39169,39009,39029,39033,39059,39071,39077,39083,39103,39107,39109,39143,39145,39157,39161,39167,39171,39173,39001,39005,39011,39013,39021,39025,39027,39031,39037,39043,39047,39051,39053,39055,39063,39065,39067,39069,39073,39075,39079,39081,39091,39097,39101,39117,39119,39123,39125,39129,39131,39133,39149,39155,39159,39175
Prices are for every matching row; the link re-counts and gives the exact price, and total=<rows> buys fewer.

See it on a map (counties shaded by count, the matching ones outlined): https://locationlists.com/find/map?areas=county&dataset=nursing-home-list&state=OH&b.dataset=hospital-list&has=a&lacks=b&in_state=OH

269 place(s) in set b could not be put in a county (no usable county, coordinates or ZIP); an area listed as having none of b could hold one of them.

Counts only, free. No rows or contact details.
JSON: https://locationlists.com/find?areas=county&dataset=nursing-home-list&state=OH&b.dataset=hospital-list&has=a&lacks=b&in_state=OH&format=json