Ok,
Lets deal with some Possibles here as you mention two email/Apple IDs
I have a very old Apple ID from 2003 based on a dial up Internet Service I had.
When iCloud started I linked to iCloud and also got an @me.com email ID in addition. Later Apple Issued an @icloud.com name based on the @me.com one.
I can log in to iCloud https://www.icloud.com/ with any of those three Apple IDs but use the same Password.
iMessages will treat each though as separate IDs...
So...
I uses one in iMessages and other in System Preferences > iCloud to sync Mail and Contacts with other devices (I tend not to use the others)
If I change the password it does so for all three "Names"/emails.
What is also possible is to Use an Apple ID linked to an iCloud account as the iMessages Account but also Add another mail in the Messages > Preferences > Accounts > iMessages account.
This could be another Apple ID but not linked to the original the way mine is.
If you changes this Password that should just carry on working the way it did before.
However if you try to make this the iMessages Apple ID then you will have issues.
The stored settings that involve using the Serial Number are linked to the Apple ID and password you were originally using.
There are situations we have seen in the past when iMessages appear out of order.
This can be related to the Apple ID or iPhone number being Off line for a while and the server not sure which ones you have had on the Mac.
The most common "fix" is to reset the NVRAM.
I would leave this though until it is clearer about which Apple ID you are using.
The Apple ID on the Mac version and the iPhone version should match to sync properly.
8:00 pm Monday; February 15, 2016
iMac 2.5Ghz i5 2011 (Mavericks 10.9)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad