If you are summing $z$-scores it seems highly likely that the result contains some negative values, so logarithmic transformation is likely to be out of the question for that reason alone.
The usual definition of $z$-scores is (value $-$ mean)/SD, so values less than the mean necessarily map to negative $z$-scores.
I am amazed that you don't report a problem here; perhaps there is some other fudge or twist you are not reporting. The only reason that might not bite is if negative $z$ scores on one variable were always counteracted by positive $z$ scores on the other, but even in such an extreme case, the rationale for adding variables has evaporated if they are so different in sign and magnitude.
Note that standardization of either variable will do nothing to remove skew. You seem surprised that the result of addition is skewed collectively, but if the individual variables were skewed, then in most cases in practice their sum will be even be more skewed.
Stepping backwards, you don't give a reason for wanting to do this in the first place. I can think of two ways forward.
Your analysis might be easier to conduct and interpret if you kept the two variables separate.
It can make perfect sense to add variables that are on the same scale, but then they have the same units, and standardization is not only not needed but also likely to lead to loss of information.
To get more specific advice, name the two variables you want to add as outcome, and list the values you regard as outcomes. Any more context you can add is likely to help.
EDIT Thanks for posting the data. For exploration I omitted individuals with any missing values. Of the rest, note that some have no values at all for some visits.
I note that
sxscore is slightly skewed, but I see no statistical reason not to analyse the values as they come.
weeklysx scores are highly skewed. That needs some careful consideration. Much although not of all the skew comes from people reporting
sxscore of 8. Drilling down shows two patients contributing six of those scores of 8. The graph shows conventional boxes for medians and quartiles, with all values shown in rank order. Thus about half the values should be inside the boxes and half outside. This is a quantile-box plot if you need a name.
It's not even evident that you should transform these
weeklysx scores. I looked at the logarithms, but they over-transform, exaggerating small differences. It seems that you have some people reporting very sick some of the time, which presumably is no surprise.
My advice in summary:
On various statistical grounds, I would definitely not mash these two outcome variables together in any way. That could not produce any measure easier to think about.
It should be evident, but standardization does nothing about skew. They also make any other transformation typically more difficult, not less, given the production of negative scores (and just possibly of zero scores).
It is not even obvious that any variable is better off transformed here.
I suggest that you consult someone with more statistical experience at your workplace.