Nitpick, one Z in Hazard. The TV show added another one for unknown reasons. (unknown to me, at least). And Hazard is the county seat of Perry County.
Since court systems are often organized by county, I think I could name quite a few.
In Kentucky, for example, Pike, Letcher, Jefferson, Harlan, Knott, Floyd, Boone, Jackson. And Muhlenberg County, from the song.
Limiting just to places I’ve lived, it occurs to me that I’ve only ever lived in one county that shared it’s name with a city therein, so that leaves:
Richland County, SC
Forsyth County, NC
Mecklenburg County, NC
Dade was renamed in 1997 or are you referring to a Dade outside of Florida?
In Georgia:
Macon is in Bibb County (actually consolidated), not Macon County.
Forsyth is in Monroe County, not Forsyth County.
Monroe is in Walton County, not Monroe County.
Plus lesser sites like:
Sumter (unincorporated) is not in Sumter County.
Houston (ditto) is not in Houston County.
None of the Peach*s are in Peach County. (E.g., Peachtree City, etc.)
When you have 159 counties, things get a little messed up.
Native Georgians would be hard pressed to recite a significant fraction of them.
You don’t need quantity to get quality confusion; the town of Wyoming PA is in Luzerne County and not Wyoming County. If the state of Wyoming is named for the town, county or region (Wyoming Valley) can be a hot debate among NE PAers. I usually just flip a coin. 
My favorite counties in Georgia are Bacon and Coffee, and my favorite thing about them is that they are adjacent.
As Gawd intended.
And Cheyenne, Wyoming, is in Laramie County, while Laramie, Wyoming, is in Albany County.
I know two Orange Counties, in California and North Carolina.
Orange: California, Florida, and I guess N. Carolina huh?
Jefferson: Well what state doesn’t have a Jefferson County? I know of at least four that do.
There is a Cleveland County in Oklahoma, but no Cleveland city
There is a Garfield County in Colorado, Oklahoma, and NJ that I know of, and I don’t know of any Garfield City anywhere (I bet there is one though)
There is a Pike County in Kentucky, and I’ll bet there are some in other places
There is a Lafayette County in Colorado and I’m betting there are about 10 more. There is also a Fayette County in Kentucky, no La.
Pontotoc County in Okla. and Mississippi.
Outside of places I’ve lived I can usually only think of Multnomah, Dutchess, Marin, and Sandia (there may be a town called Multnomah…) I’m sure that I could think of more if confronted with a clue in a crossword puzzle although it’s usually Taos in that case.
I used to drive through four counties on my way to work: Denver, Arapahoe, Douglas, and Jefferson although my actual route went Denver-Arapahoe-Denver-Jefferson-Douglas-Jefferson because of weird boundaries. There was an alternate route where I could go straight to Jefferson but it took longer.
The only ones outside of California that pop into my head are Clark County, Nevada (Las Vegas) and Fulton and Cobb County, Georgia (the former from Fulton County Stadium; the latter thanks to the late Ray “Big Boss Man / Big Bubba Rogers” Traylor).
Which Orange? There is a city of Orange in Orange County, California.
I’m pretty sure I’ve been in every county in Maryland, if only driving thru. I lived in King George, VA, so I know several counties in that state. Lived in NE Florida for a lot of years, so I know a bunch of counties there. But I don’t know that I paid attention to county names in the various places I was stationed while in the Navy. I’d need to look them up to jog my memory.
The town of Alameda is in Alameda County.
The town of Camden is in Camden County.
Cape May Courthouse is in Cape May County.
Salem is in Salem County.
Burlington is in Burlington County.
Cumberland is in Cumberland County.
Middlesex is in Middlesex County.
Passaic is in Passaic County.
Sussex is in Sussex County.
And Texas, Virginia and New York. The latter got famous a bit back due to a “reality” TV show.
Note that New York doesn’t grow oranges, but William (III) of Orange and all that.
As noted before, there is 1 Peach County. Two Lemon Counties. Guess how many Apple Counties there are in the USofA?Zip. Nada. Sorry, Mom, no pie.
Let’s not forget Wanker County, Wisconsin, ancestral home of Peggy Bundy.
Douglas is the seat of Coffee county, not of Douglas county. When I moved there in '84, I found Georgia is a very county-centric state. I swear they make elementary school kids memorize all 159 of the suckers. Ask someone from Georgia where they live, and they’re more likely to say the county than the city.
The only county thing I remember from my youth in Indiana was the license plate game. Back then, the counties were numbered alphabetically, and the plates started with the county number. 18 was Delaware, 48 was Madison, 49 Marion county, and so on. Folks would read the numbers and try to name the county. Lists were given out at various places for this purpose. Those numbers are still on the plates but now they’re small and down in one corner so they’re usually illegible on the road.
The list of counties west of the Rockies is too long and boring so:
New York-the five boroughs (and I know Brooklyn is Kings County)!, Suffolk, Nassau, Westchester, Orange
Maryland-Prince Georges
Ohio-Cuyahoga
Illinois-Cook
Florida-Pinellas, Dade
Minnesota-Hennepin
South Dakota-Pennington (Rapid City)
Bad habits followed the name west.
I grew up in Maryland, and applied to teach in every county.
King County, WA
Allen, Steuben, and DeKalb counties in NE Indiana
Collier County, FL
Beaver, Texas, and Cimarron Counties in the Oklahoma Panhandle
In Arizona, most of Apache County is on the Navajo reservation.
I’m Cuyahoga born. Did anyone mention Geauga County? Lake County?
I’m in Fairfax County, which counts too. The town of Fairfax is independent under the Virginia constitution, so it isn’t part of Fairfax County. There are also Loudoun, Prince William, and Fauquier nearby.