The important part to add here is that a carbon tax is still a tax, which pays for stuff like streets, schools and all sorts of other government services. So it is not money wasted. It also hits rich people harder then poor ones.

Then about carbon credits, they are traded, but that is part of the idea. Basically the supply is supposed to limited. For badly designed schemes that might not be the case due to too easy offsets, but for example the EU scheme has no offsets. The problem a lot of green technologies have is that they are more expensive then fossil fuels, if you exclude damages caused by fossil fuels. So if a carbon credit costs more then that difference people will choose the green process. By having this trade freely and force a large part of an economy to have to buy them for emissions, the idea is the cheapest green alternatives are adopted first and emissions go down that way. Then the number of credits and permitted emissions go down and the next green alternative gets adopted and so forth until you do not have any fossil fuels left anymore. The problem is that a lot of green technologies are not cheap. So there is potential to really hurt the poor with it. However it is a good way of having a controlled phase out of fossil fuels and with subsidies you can support the poors green transition.