Unless there is some specific reason for people being NA, and unless you are interested in that reason, then I would say to not include people who are missing.
You don't need an exact test here; all the cell sizes are reasonable.
However 1) Don't you want some form of regression instead? and 2) Why is Albumin dichotomized into low and high? Dichotomizing continuous variables is usually a bad idea (see Royston, Altman & Sauerbrei).
If you have actual values for albumin, I suggest a linear regression albumin~case, possibly with other covariates added, if you have data. This is especially important if this is an observational study, but is still worthwhile if it is an experimental one, because covariates can vary between groups, even if assignment is random, and because covariates can affect other regressors.
How to do this has been elaborated by Rubin (1988, p.87) and Li et al. (1991).
First, you take mean $\chi^2$ across multiply imputed data sets.
$$\chi^2=\frac{1}{m}\sum _{l=1}^{m}\chi^2_l$$
You estimate the relative increase in variance:
$$r=(1+\frac{1}{m})\frac{1}{m-1}\sum _{l=1}^{m}(\sqrt{\chi^2_l}-\sqrt{\chi^2})^2$$
And the test statistic:
$$D_x=\frac{\frac{\chi^2}{k}-\frac{m+1}{m-1}r}{1+r}$$
where $k$ is the degrees of freedom of $\chi^2$. Now, $D_x$ is F-distributed:
$$P_x = Pr(F_{k,v} \gt D_x)$$
where
$$v=k^{-3/m}(m-1)(1+\frac{1}{r})^2$$
which gives you the Chi-square test across multiply imputed data sets.
Li, K.-H., Meng, X.-L., Raghunathan, T. E., & Rubin, D. B. (1991). Significance levels from repeated p-values with multiply-imputed data. Statistica Sinica, 1(1), 65–92.
Rubin, D. B. (1988). Multiple Imputation for Nonresponse in Surveys. John Wiley & Sons.
Best Answer
What I would do to assess the type of missingness is partition your results into four groups: complete cases, first answer blank, second answer blank, both answers blank.
In complete case set, look at the marginal distributions of $X_1$ and $X_2$, separately. Compare the marginal distribution of $X_1$ in the complete case category to the marginal distribution of $X_1$ where the second answer was blank. See if they're similar. (You could do some test if you'd like - perhaps chi-squared to compare the categories under missingness and completeness - if you want to more methodically make a decision.)
Repeat for $X_2$. If the distributions are similar, then it's evidence that your data are MCAR (missing completely at random) as missingness doesn't affect the marginal distributions. This will help you to assess what mission data method (likely imputation) you should use.
Hope this helps! Let me know if this is unclear.