The problem was fixed after following an idea from post, which talked about how older versions of ANSYS require different configuration compared to newer versions. The new license server had port 2325 closed. But because I was running an older version (2019), the client needed that port opened along with the ansymld.ini variable ANSYSLI_SERVERS set for that port. So, after the license server opened up that port, and after changing the client's ansyslmd.ini to the below, we can run Fluent and Structures again:
SERVER=1055@licserverhostname
ANSYSLI_SERVERS=2325@licserverhostname
Running ansysli_util -statli $PORT@$HOSTNAME was helpful as a quick test showing that the license interaction was good.
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