$$\int_0^\infty \sqrt{x}\,\mathrm{Erfc}(x+a)\,\mathrm{d}x\rightarrow \frac{e^{-a^2}}{(2a)^{5/2}},\;\;\text{for}\;\;a\rightarrow \infty,$$ so $C=2^{-5/2}.$
I obtained this asymptotics by integrating the large-$a$ expansion of the error-function, $$\text{Erfc}\,(x+a)\rightarrow \frac{e^{-(a+x)^2}}{\sqrt{\pi } a},\;\;\text{for}\;\;a\rightarrow \infty.$$

$$\int_0^\infty \sqrt{x}\,\mathrm{Erfc}(x+a)\,\mathrm{d}x\rightarrow \frac{e^{-a^2}}{(2a)^{5/2}},\;\;\text{for}\;\;a\rightarrow \infty,$$ so $C=2^{-5/2}.$ Corrections are smaller by a factor $1/a$.
I obtained this asymptotics by integrating the large-$a$ expansion of the error-function, $$\text{Erfc}\,(x+a)\rightarrow \frac{e^{-(a+x)^2}}{\sqrt{\pi } a},\;\;\text{for}\;\;a\rightarrow \infty.$$