I managed to get it working.
Since the problem seemed to be that libjawt.so was missing, I did the obvious thing:
[root@localhost ~]# dnf install libjawt.so
Affterwards, when I tried launching the java file, I still had the exact same error. Looked like libjawt.so wasn't in whatever path java was searching through, so I had to do this as well:
[user@localhost ~]$ sudo find / -name 'libjawt.so' -ls 2>/dev/null
951358 8 -rwxr-xr-x 1 root root 6272 Oct 20 04:33 /usr/lib/jvm/java-1.8.0-openjdk-1.8.0.111-1.b16.fc25.i386-debug/jre/lib/i386/libjawt.so
271809 8 -rwxr-xr-x 1 root root 6928 Oct 20 05:04 /usr/lib/jvm/java-1.8.0-openjdk-1.8.0.111-1.b16.fc25.x86_64-debug/jre/lib/amd64/libjawt.so
[user@localhost ~]$ echo $LD_LIBRARY_PATH
(empty line)
[user@localhost ~]$ export LD_LIBRARY_PATH="/usr/lib/jvm/java-1.8.0-openjdk-1.8.0.111-1.b16.fc25.x86_64-debug/jre/lib/amd64"
[user@localhost ~]$ java -jar Downloads/Tales\ of\ Androgyny\ Linux64\ v0.1.15.1/TalesOfAndrogyny.jar [user@localhost ~]$
And now I can launch the file just fine.