Usual procedure - update the shared library, restart affected services. Go - recompile everything.
What's the issue with that?
On Windows you have to package most shared libraries with your app anyway so you have to get a new version of it anyway.
... look for programs to break at runtime because of some unrelated API change in the shared library.
shouldn't
When the patched library is not part of Debian Stable or RHEL's repositories (for example, if you require features from a release less than a year old) all bets of API stability are off.
OpenSSL and libc are not the only libraries which are patched for security that people use.
In fact, what about just switching connections to freshly launched VMs?