The Soyuz-class appears to have been built as a variant of the Miranda-class. The differences in the classes are an enlarged aft cargo and shuttlebay section, as well as several large outboard sensor pods. The Soyuz-class was retired in 2288.

The USS Bozeman is mentioned in Generations and First Contact, so obviously there's still one Soyuz-class ship in service. Unless it was decommissioned and another ship was given the name.

Why the Soyuz was retired so early in its life is unknown, of course the year that it entered service is also unknown so it's possible this class existed before the Miranda.

According to the book, Star Trek: The Continuing Mission - A Tenth Anniversary Tribute to The Next Generation, in the Bozeman concept drawing the registry was originally USS Courageous NCC-1861.

It was originally hoped that a new design could be developed for the Bozeman, but budget constraints dictated the reworking of the existing USS Reliant model. The modifications were designed by Greg Jein and Mike Okuda.

The class was named for the Russian spacecraft that shuttled cosmonauts up to the Salyut and Mir space stations.

USS Bozeman NCC-1941
   While commanded by Captain Morgan Bateman the Bozeman disappeared near the Typhon Expanse in 2278 where it remained until    2368. During those 90 years the crew did not age or were unaware of the passage of time. The Bozeman was one of the 15 ships sent    to the Federation side of the Neutral Zone to investigate the Romulan military buildup. The Bozeman was also in the armada to    intercept the Borg incursion to Sector 001 in 2372.
   Its registry is a homage to the Steven Spielberg movie 1941. Named for the Montana city.
   "Cause and Effect" - TNG. "All Good Things…" - TNG. "Star Trek: Generations" "Star Trek: First Contact"

Type: Cruiser
Length: 243m
Speed: 8.8
Armament: Six Type 6 phasers
Two torpedo launchers