Such problems can be solved by Laplace's method. The starting point is the observation, that the values of the terms of the sums are unimodal as function of $n$ for fixed (large) $x$, i.e., have one maximum. If the maximum (as in the given cases) is sharp enough, the terms of the sum can be approximated by a Gaussian. This Gaussian is then integrated on the whole $n$-axis, where $n$ is considered as a real number. The integral often converges. The contributions from negative $n$ are negligible.

The maximum, $n_{0}$, for $\frac{x^{2 n +1}}{n!\sqrt{n}}$ for fixed $x$ can be evaluated to
$$
n_{0} = x^2-1-\frac{5}{12}x^{-2}-\frac{1}{2} x^{-4}-\frac{123}{160} x^{-6}+\frac{359}{180} x^{-8} + O(x^{-10}).
$$
The sum is approximated by an integral over a Gaussian (which we get when expanding around $n=n_{0}$ to second order in $\tau$)
$$
C(x) \approx \int_{-\infty}^{\infty} d\tau\ \exp\left( \ln\left(\frac{x^{2 n +1}}{n!\sqrt{n}}\right)|_{n\rightarrow n_{0}+\tau}\right),
$$
consistently expanded to $O(x^{-10})$ after integration. The result is
$$
C(x)= e^{x^2}\left(1+\frac{5}{12} x^{-2}+\frac{157}{288} x^{-4}+\frac{49729}{51840} x^{-6}+\frac{417001}{497664} x^{-8}+O(x^{-10})\right).
$$
I could not find any systematic for the coefficients. Only the numerators of the higher order terms seem to contain rather large prime factors.

The result is different from your findings, though. However, numerical evidence suggests that the above asymptotic expansion is correct. 

My result for $L(x)$ is
$$
L(x)=e^{x^2}\left(x^{-2}+\frac{23}{12} x^{-4}+\frac{1525}{288} x^{-6}+\frac{949099}{51840} x^{-8}+O(x^{-10})\right),
$$
with again rather large prime factors of the nominators of the higher order coefficients. The maximum of the terms as function of $n$ is reached (up to $O(x^{-10})$) for $n=n_{0}$ with
$$
n_{0}=x^2 -2-\frac{23}{12}x^{-2}-5 x^{-4}-\frac{2643}{160}x^{-6}-\frac{3007}{45} x^{-8}+O(x^{-10}).
$$
A numerical test shows very good quality of the asymptotic expansions for $C(x)$ and $L(x)$, even for $x$ around $2$.



All calculations where done with Mathematica 11.

Edit: I corrected a typo in the coefficient of $x^{-8}$ in the expansion of $n_{0}$ for the asymptotics of $C(x)$.