Is there anyway to add another mailing address to my apple id?

If you mean your billing address, you change / update your billing address:

(macOS) Set up your Apple ID payment method and shipping address on Mac

Set up your Apple ID payment method and shipping address on Mac – Apple Support (UK)

or 

Sign in to your Apple ID account page and edit / update your Apple ID details -> https://appleid.apple.com

• edit your billing addres.

or 

To change the billing address you can do (macOS & Windows OS) via iTunes

Start Music (-> Click on iTunes) -> side bar click iTunes Store -> go to Account (top bar) -> View Account -> Sign in -> Apple ID Summary -> Billing Address -> Edit.

or 

(iPhone & iPad) Go to -> Settings -> tap your Profile -> Payment and shipping -> Change Address.

Edit the Address -> tap Done.

Go to Apple ID - Official Apple Support>Visit your account page>Payment & Shipping>and change your Shipping Address.

Yes, see what was posted.

On the day or a day after you move, you change your billing address.

I think I will try and change my mailing address on Tuesday and Wednesday

Basically, it doesn't matter that much, you can change it on Tuesday or on Wednesday.