If I marry someone in a different service after having already received my orders, am I able to change my orders to get a joint assignment?

Each service has a married couples program for couples WITHIN their respective service (the Married Army Couples Program, for example). Sadly, there is no overarching joint program between all the different branches.
The level of logistical complexity in making that a reality somewhat precludes it's existence, unfortunately.

You need to get married and enroll. Be prepared for bad news. There is a developed topic page for this topic here on RP: https://www.rallypoint.com/topics/marriage/about

It has resources. Understand you are already an inbound to that installation, if not a specific BDE. Not sure what LT Asssignments branch can do. You need to do some leg work to figure out assignments that will work for both of you. You may need to suck down that assignment to Stewart while your spouse to be gets their initial assignment with the aim of the follow on being Joint domicile. Especially as you are not yet married. There is no assignment consideration for either of you without being married.

Hard reality here is being married as officers to another branch will be very difficult for you to both be active duty and reside close to each other. You getting married and him having no orders really wont change your orders because he does not have orders. Kinda hard to change your orders to a location that he does not have yet. Is that the question you were really trying to ask?