Wolverine's Adamantium did not degrade from true Adamantium to secondary Adamantium. Adamantium is too strong and no super-charged healing power in the Marvel universe can alter the molecular structure of the strongest material in the Marvel universe. Secondary Adamantium is a far less expensive form of Adamantium that is produced to be used in mass quantity. While secondary Adamantium is nowhere near as strong as true Adamantium, it is still far stronger than any other steel or titanium alloy produced.
Wolverine's bonded Adamantium became Adamantium beta. "Adamantium Beta" is the result of the healing factor finding a way for Adamantium to allow Wolverine's bones to continue to function as normally as a part of his body. Bone marrow produces red blood cells so if the Adamantium completely closed off the bone from the rest of the body, Wolverine could never produce blood again. In reality, the Adamantium became a part of his body and actually helps regulate/control and sometimes enhance his healing factor and abilities. Wolverine can take punches from others with "Hulk"-level strength without having bones broken.
Comparing Adamantium against Vibranium is very difficult because the properties of each metal are so different. Vibranium is found in nature in the Marvel universe while Adamantium is an artificially produced metal alloy/resin. Vibranium can absorb and redirect forms of energy, making it very strong and resilient to damage in addition to being very tough already. Adamantium is physically impervious to most physical, chemical, and biological harm but it does absorb any energy. On equal terms, Adamantium can scratch and even cut Wakandan Vibranium, but it will probably require a very great amount of force to do so. Antarctic Vibranium would damage Adamantium because it destabilizes the molecular structure of all metals. Adamantium would probably have to be used to cut through super quick if possible.
In relation to Captain America's shield, Adamantium cannot cut through the shield because the shield is made of the original metal that was reverse engineered to create Adamantium alloyed with Vibranium to produce a metal that is stronger than true Adamantium. If the shield was made with only Vibranium, Wolverine's claws could probably cut through the shield with a lot more force than even Wolverine can produce. You would probably need someone with Adamantium claws and Hulk-level strength (now that would be one scary being in the Marvel universe).