Got multiple accounts connected for the same carrier? Not using your current Primary account as much? Make one of your other accounts the Primary account for that carrier. Your entire shipping team should agree to this change, since it will affect all users in your ShipStation account. That aside, the 'promotion' process is really simple.
Head to your Account Settings (the gear icon in the upper right).
Next, select Shipping and then Carriers & Fulfillment from the sidebar on the left.
Once there, click the "Edit" link for the shipping account that you'd like to promote to Primary status.
This can vary from carrier to carrier, but look for an option along the lines of "Make this my primary account."
Select that option and confirm any changes as needed. This also 'demotes' whichever account used to be your Primary account.
An order that wasn't ever assigned to use a specific shipping account will simply use the current Primary account whenever you ship that order.
Orders assigned to use specific shipping accounts won't have their selections change if you promote to or demote from Primary status. In these cases, the "(Primary)" indicator will simply show up or go away as appropriate.