Sendmail sends a hostname with the HELO command to the other side when making a connection. But it is not necessary that this hostname matches one of your domains. It it mainly an identifier of the machine sending out the mail, rather than the domain which is used in the From: address. To avoid rejection while sending emails, you should make sure that this HELO hostname returns the IP address of your server if the receiving mail server performs a DNS query on it. I.e. if your sendmail sends out smtp.example.com during the HELO command, be sure that smtp.example.com returns the IP address of you server.
The second check often done--and required by the SMTP specification--is that the IP address of a sending mail server is checked for the existence of a reverse PTR record. The standard requires the existence of a PTR record, but it is not necessary that this matches the domain name sent during the HELO command or the domain name which is associated to the sender's email address.
You should therefore be able to send email from several different domains from one email server without running into problems with mismatches between your From: domain name and your server name.
If you want to increase the chances that your email arrives properly, you can implement SPF or DKIM. Both are technologies which give receiving mail servers more information to judge if emails are legitimate. SPF records protect the bounce address of an email and are practically only saying to receiving email servers that your email server with that specific IP is allowed to send emails on behalf of your bounce address domain. It protects your bounce address domain from back scatter caused by bounced spam messages. SPF doesn't say anything about the From: address. DKIM on the other hand sends an encrypted hash as a header in the message which validate the contents.