Usually, Pigouvian taxes like the one OP was proposing are calculated to cover the complete externalized costs of the item being taxed. So in this case, it would include the costs of adjusting to climate change. This would make alternatives relatively cheaper, as well as raising revenue to support research and construction of new energy infrastructure, as well as point addressing of specific projects identified to reduce the impacts of climate change. Of course, this is all if you believe that the current political climate (not only in the US, but in large middle income economies as well) can support such taxes. And then you have to decide what the tax should be! Ultimately, cap and trade turn out to be far easier.

one benefit of pigovian taxes is increased revenue to ameliorate the externality, but it's not the main goal of the tax. the cost itself reduces demand to the socially optimal level. the government could well burn the tax revenue (or in this case, recycle the paper) and still achieve the objectives of the tax.

If you can convince politicians to make the tax, it's trivial in comparison to earmark all the money for carbon capture and carbon capture research. (And it won't matter if existing funding gets cancelled because the influx would be a thousand times bigger.)

Transfer the money directly into public transit. Or construction of renewable energy. Or towards education or health care.

I mean you can just put it in the general fund as a way of discouraging use, like you said. I think that would work just fine.

What? Are you fking serious? We have to invest tens of trillions in scalable carbon sequestration right now or there's no future for anyone except to starve or be part of another world war.