Methane breaks down into CO2 and water in a few years, which is why half life of methane is relevant. My understanding is that for CO2, thinking in terms of half-life is less relevant, the CO2 goes to plants and to the ocean, and if those two sinks are at saturation, then CO2 levels in the atmosphere will keep increasing.

But there's a good article I found now which might shed more light : https://euanmearns.com/the-half-life-of-co2-in-earths-atmosp...

Some amount of carbon is sequestered by plants getting buried when they die, though not as much as in the past because we have well- slated decay organisms in the biosphere. Some makes its way into the sea and gets stuck there. And a decent chunk is bound up as carbonate minerals from the weathering of the Earth (one of the future biosphere death states is when this process outstrips CO2 production due to the increase in the sun's luminosity about 700 million years from now).

When it rains, Co2 in atmosphere leaches the calcium in rocks forming calcium carbonate that runs off to the sea eventually. Hundreds of millions of years from now, our actions leave their mark on some ancient seabed, as a thick white stone(like Dover, England) . The higher the Co2 in the atmosphere, the more it rains, because more water evaporates due to higher temperature, causing faster reduction of Co2 in the atmosphere. Thus there is a natural balance in the cycle. Unfortunately for us, it is way slower than the speed with which we pumped the Co2 out. It takes tens of millions of years to get back to preindustrial levels (of < 200 ppm) Co2