MX Records on GoDaddy that point to GoogleApps

MX Records on GoDaddy that point to GoogleApps

My company has a domain that is hosted on amazon, and the domain name
bought from GoDaddy, and the required NS are set on GoDaddy.
When we were developing the site, we needed to send emails, and used
GoogleApps with the required MX Records set on GoDaddy to send and receive
emails through GMail (this feature still is active though the site is
finished). We have 2 email accounts set in GoogleApps Admin console
through which we send emails.
We would like to send emails through other accounts, that will not be set
in GoogleApps Admin Console because we would have to pay for each
account/month. For this we've setup and email server, and some email
accounts (with @ourdomain.com).
Sending emails from this accounts to accounts on the same domain works,
also receiving emails from the same domain. Sending emails from this
account to other domains like google.com or yahoo.com also works.
Our problem is that the new created accounts (in our mail server) do not
receive emails from other domains. Ex: sending an email from my gmail
account to me@mydomain account is not working and I receive the following
header:
"Delivery to the following recipient failed permanently:
my_new_account @ mydomain.com
Technical details of permanent failure: Google tried to deliver your
message, but it was rejected by the server for the recipient domain
mydomain.com by aspmx.l.google.com. [2607:f8b0:4003:c02::1a].
The error that the other server returned was: 550-5.1.1 The email account
that you tried to reach does not exist. Please try 550-5.1.1
double-checking the recipient's email address for typos or 550-5.1.1
unnecessary spaces. "
So if Google can't find my email address setup on the other email server,
cand I make some changes so that is does? Or does anyone has other
solution to keep Gmail for those 2 accounts and user the other server for
the other accounts?