Series explanation, it was after ST3 and all they had at the time was the Oberth, Miranda and Excelssior ships, so, tits up ship of the week was usually one of those. Back then it was quite expensive to make a ship, 100,000$ and up, so they tried to get there milage out of them. We eventually get the Nebula, Ambassador, Vorcha, Negvar, Pastuer before the advent of CGI. So I don't fault them for reuse, its what they had at the time, and the budget for. Enterprise mostly solved that with new ships poping up most episodes, though even then there was some reuse.
In show Explanation: Excelsior class is 80 years old by the time of the start of Tng, they keep them updated, but new ships are the front line of exploration. So by this time, excelsior are used mostly for internal transportation. Admirals, dignitary, cargo, colony work, border patrol,etc. busy work.
For admirals, it depends who they are, where there going. Someone like Admiral Hansen probably came from Earth, there are probably a number of ships for Earth Defense that get rotated around, and have ships that are "Ready to go" to back up near Earth ships in an emergency. So Admiral Hansen probably called up space dock, asked for the next ship in the rotation, and hopped on board. Thats someone in a hurry.
Now for Admirals that aren't in any hurry, then they grab whatever ship happens to be going in there direction. Kinda like standby! Even TOS had an admiral/commodore/dignitary on board that was just hitching a ride.
For Admirals that are in charge of star bases or sectors, there's probably a "Flagship" that is stationed by the base, that he can hop on and go where ever he's needed.
In the real world, Admirals are in charge of fleets, example is a carrier task force. in the aircraft carrier, there's an admiral.
In the books, the Full Circle Fleet has an admiral in charge, so if theirs a task force, or large enough group of ships going out, there's probably an admiral somewhere in there.