One more bit of info. After leaving s_client up for a bit, I got another line

CONNECTED(00000003)
675481056:error:140790E5:SSL routines:ssl23_write:ssl handshake
failure:/usr/src/crypto/openssl/ssl/s23_lib.c:177:
---
no peer certificate available
---
No client certificate CA names sent
---
SSL handshake has read 0 bytes and written 307 bytes
---
New, (NONE), Cipher is (NONE)
Secure Renegotiation IS NOT supported
Compression: NONE
Expansion: NONE
No ALPN negotiated
SSL-Session:
    Protocol  : TLSv1.2
    Cipher    : 0000
    Session-ID:
    Session-ID-ctx:
    Master-Key:
    Key-Arg   : None
    PSK identity: None
    PSK identity hint: None
    SRP username: None
    Start Time: 1557988789
    Timeout   : 300 (sec)
    Verify return code: 0 (ok)
---

I guess I wasn't patient enough before. But the certs are in the exact
same place as they've been all along and are working fine with apache.
I can also send mail, if that is at all relevant. (postconf -n is in
another post in this thread)