Does Adamantium exist in the MCU

Adamantium and Vibranium vulnerabilities

First things first: Adamantium has a slightly more complicated history than just becoming a metal or finding a metal one day. The first "version" of Adamantium was Proto-Adamantium, which Dr. Accidentally discovered by Myron MacLain during World War II. It was then recreated in the 1960s, but using a combination of resin as opposed to the vibranium in the proto-adamantium. You can find more about adamantium at the end of this answer 1

In a metallurgical accident, MacLain created the indestructible vibranium-steel compound from which the shield of the super soldier Captain America was made. MacLain spent decades duplicating the process, and while unsuccessful, he created True Adamantium instead in the 1960s.
Adamantium -

Vibranium has a simpler history and exists as a naturally occurring meteoric metal that, although rare, occurs naturally in the universe. Most of the earth's vibranium exists exclusively in Wakanda.

Now to the question at hand. Using the example of a magic that broke Captain America's shield, I found an iron-vibranium alloy. It was destroyed by King Thor in Thor Vol. 2 # 73. Thor, wielding the power of the Odinforce, destroys the shield and kills Captain America. However, the timeline is restored in # 79. In the same comic published just a few panels before, Thor melts Wolverine's Adamantium Beta with the Odin power.

(Click on all of the images to enlarge them.)

On another earth, where adamantium is classified as weaker, Valkyrie smashes Cap's shield with a magical sword under the power of the sorceress. (Ultimate Comics Vol. 1 New Ultimate # 2)

1 This metallurgical casualty was shaped into a disk and used to make Captain America's shield. As noted above, MacLain continued to attempt to restore the "Proto-Adamantium" but was unsuccessful. Instead, true adamantium was discovered in the 1960s and was to become the standard for adamantium. While Proto-Adamantium was formed using steel, vibranium, and an unknown catalyst, True Adamantium is created by mixing certain chemical resins, the exact composition of which is closely guarded government secret and exists in a malleable state for 8 minutes before hardening regardless of temperature. The only known substance that can penetrate adamantium is the compound known as Antarctic Vibranium, also called "anti-metal".

There is also secondary adamantium, which is far weaker and can be broken by people with superhuman strength.

Because of the prohibitive cost of creating Adamantium, many parties have resorted to the use of a somewhat weaker compound called Secondary Adamantium, which was once used to coat the sentient computer FAUST.
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Adamantium Beta, which was formed after True Adamantium fused with Wolverine's bone and was altered at the molecular level by its healing factor:

Wolverine's mutated healing factor allowed him to survive the process and induce a molecular change in the metal that converted it to an entirely new metal called adamantium beta, which does not inhibit the biological processes of the bone.

There are also different variations for Adamantium for different earths, such as Earth-1610 (Ultimate Earth) which is not as strong as that of Earth-616 etc. Finally the Soviet Union tried to create Adamantium as mentioned in the link above, but were not exactly successful:

Attempts by the former Solviet Union to reproduce the metal resulted in the formation of carbonadium, a weaker but far more malleable form that was used to make retractable coils for Russian super soldier Omega Red.