It seems like this is really your main question:
Why does adding more $\mathrm{CO_2}$ to the atmosphere increase the greenhouse effect if the atmosphere is already opaque in the absorption bands of $\mathrm{CO_2}$?
This is a good question. The main reason that the atmosphere is close to opaque in the strongest absorption bands of $\mathrm{CO_2}$ is of course the absorption by $\mathrm{CO_2}$ itself in these bands. The absorption bands are said to be saturated. This means that any added $\mathrm{CO_2}$ does not absorb as much radiation extra as if the strongest bands were not saturated, but it does not mean that they don't absorb any extra radiation.
The infrared absorption cross section of a gas can be seen as a superposition of a number of absorption lines. Each line has the spectral shape of a <PERSON> profile (really a <PERSON> profile, but the difference only matters for the upper atmosphere) and gives a contribution $\sigma_i$ to the full cross section:
$$ \sigma_i(\nu)=\frac{S_i\alpha_i}{\pi}\frac{1}{(\nu-\nu_i)^2+\alpha_i^2}, $$
where $\nu$ is the wavenumber, $S_i$ is the line strength, $\nu_i$ is the line center wavenumber and $\alpha_i$ is the half-width at half maximum of the line. Now, consider the transmission of a gas with only a single absorption line (assume constant temperature and pressure). The Transmission $T(\nu)$ is given by the <PERSON>-Lambert law:
$$ T(\nu)=e^{-C\sigma_i(\nu)}, $$
where $C$ is the column of the gas. 439
Below I have plotted the transmission for columns of very different magnitudes:
When the column is small, the amount of radiation absorbed is approximately proportional to the column $C$, since the Beer-Lambert law can be linearized to: $T(\nu)\approx1-C\sigma_i(\nu)$. But when the line becomes saturated, the amount of absorbed light instead grows mainly because of broadening of the band of almost full absorption. For heavily saturated lines, the total absorbed light is instead approximately proportional to width of this band which in turn is approximately proportional to the square root of $C$, since the denominator in $\sigma_i(\nu)$ grows with the square of the distance from the center wavenumber. This can be seen in that the width of the absorption line approximately doubles as the $C$ is increased by a factor of $4$ in the figure above. This phenomenon is sometimes referred to as square root absorption.
In reality it is of course complicated by the fact that each gas has a large number of lines that interfere with each other, and also with the absorption of other gases, but the square root approximation often holds for strong absorbers despite this. Hence the last ton of $\mathrm{CO_2}$ emitted may not contribute as much as the first, but it still contributes since the square root increases monotonously.
But it is also more complicated than this
It is not just the total opacity of the atmosphere that matters. When the $\mathrm{CO_2}$ concentration increases, the infrared radiation from the surface that is absorbed will also be absorbed at a lower altitude than before and this has other effects that someone else is probably more qualified to discuss than me. 795