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Sorry to not help much with this exact problem. I should know about it since I build my own servers from source code. But I have infrequently had to deal with the issues that hosted servers deal with.
One solution if at all possible is to use the SMTP Auth method against a server that will allow outbound email once authenticated. This server can be remote but has to also protect itself from spam relay.
Your one address is working because it is in the allowed list of domains your host server is set-up for. The originating nobody@, apache@, etc is failing at the server and can also fail at the remote server if the name/domain can't be validated.
This is a problem that will get much worse in short order for many other people. Blame the spammers for this trend.