moveonly: factor out headers processing into separate function

Summary:
ProcessMessages will now return earlier when processing headers
messages, rather than continuing on (and do nothing).

This is part of Core PR11568

Depends on D1764

Test Plan:

make check

Reviewers: #bitcoin_abc, schancel

Reviewed By: #bitcoin_abc, schancel

Subscribers: teamcity

Differential Revision: https://reviews.bitcoinabc.org/D1777