I recently ran into the same situation while comparing the xspec models bbody and bbodyrad. I'm guessing that you're taking some of these formulae from <PERSON> and <PERSON>, since you're using their notations. I will refer to these equations with RL#__, for <PERSON> and <PERSON> copyright 2004 version.
Just under RL#1.38, the text states: $I_{\nu{}}=B_{\nu{}}$, since the Planck function $B_{\nu{}}$ is the thermal radiation in an optically thick medium. 469
Back in $\S{}\S{}$1.3, the properties of $I_{\nu{}}$ are discussed, and we see that the units are erg$~$s$^{-1}$$~$cm$^{-2}$$~$ster$^{-1}$$~$Hz$^{-1}$, and that to get to flux we must integrate over solid angles, which you did and R&L do in RL#1.13:$~F=\pi{}B\left(\frac{R}{D}\right)^{2}$.
But we should be careful, as this is still dependent on $\nu{}$! If we want to get the units right, the formula should really look like: $dF_{\nu{}}=\pi{}B_{\nu{}}\left(\frac{R}{D}\right)^{2}d\nu{}$ with units on both sides erg$~$s$^{-1}$$~$cm$^{-2}$. Now we're talking.
First convert the xspec model to our $\nu{}$-dependent formulation.
$A(E)=K\times{}1.0344\times{}10^{-3}\frac{E^{2}dE}{e^{E/kT}-1} = \left(\frac{R_{km}}{D_{10}}\right)^{2}\times{}1.0344\times{}10^{-3}\frac{E^{2}dE}{e^{E/kT}-1}$ $A(\nu{})=\left(\frac{R_{km}}{D_{10}}\right)^{2}\times{}1.0344\times{}10^{-3}\frac{(h\nu{})^{2}hd\nu{}}{e^{h\nu{}/kT}-1}$ where we sub. $E=h\nu{}$
In cgs, the conversion to radius in cm and distance in cm is $\frac{R_{km}}{D_{10}} = \frac{3.086\times{}10^{22}}{10^{5}}\times{}\frac{R}{D}$. Now
$A(\nu{})=\left(\frac{R_{km}}{D_{10}}\right)^{2}\times{}1.0344\times{}10^{-3}h^{3}\frac{\nu{}^{2}d\nu{}}{e^{h\nu{}/kT}-1}=2.9\times{}10^{-47}\left(\frac{R}{D}\right)^{2}\times{}\frac{\nu{}^{2}d\nu{}}{e^{h\nu{}/kT}-1}$
Comparing this to the R&L version:
$dF_{\nu{}}=\left(\frac{2\pi{}h}{c^{2}}\right)\left(\frac{R}{D}\right)^{2}\frac{\nu{}^{3}d\nu{}}{e^{h\nu{}/kT}-1}= 4.6\times{}10^{-47}\left(\frac{R}{D}\right)^{2}\frac{\nu{}^{3}d\nu{}}{e^{h\nu{}/kT}-1}$
We get a similar coefficient, however the qualitative form is off by a power of $\nu{}$. 374