I don't think you can disable it because their increasingly stringent security algorithms impose it as a kind of de facto "two factor authentication" ("2FA") in the absence of you specifying some other form of 2FA.
I would go into your email settings on both services and specifically setup 2FA using a non-SMS based method. Select the option to use an OTP ("One-Time Password") authenticator app, such as
Microsoft Authenticator or
Google Authenticator instead of SMS messages. That way, the next time this happens the service should direct you to use the authenticator app in lieu of having a code texted to you.
These apps use time-based One Time Passwords, which automatically change every 30 seconds.* Once it's configured, when you try to login to a 2FA-protected account you first execute the first factor verification (typically, a login password) like you usually would, after which it will present a second screen that basically says, "open your authenticator app, read the 6-digit code it displays, and type it in here." Functionally, it's the same as having a code texted to you, but doesn't require you to have a connection to any cell or data service to use it.
Not only are authenticator apps more secure than SMS (which can be intercepted), they're easier to use because you don't have to wait for the text to come in, and you can use the app even when you don't have a cell connection.
To setup an account, you step through the account provider's (Microsoft, Google, et al) setup procedure on your computer, which at some point will display a QR code containing a random key. You open the authenticator app on your phone and select the option to add an account, at which point it will prompt you to point the camera at the QR code. The app will read the QR code's key and give you a 6-digit code to type back into the computer as confirmation, verifying to both sides that they are using the same secret key. From that point forward, you only need to open the app and it will display the ever-changing 6-digit code.
(Note in your case this means you'd still have to take your phone with you, even if you won't be able to make calls with it.)
Note each account you setup will use a different secret key, so you'll get a different 6-digit code for each account, and those codes change every 30 seconds.
For the sake of convenience, note many account login pages have a checkbox you can tick to say, "don't ask me again on this computer"--which means they'll skip further 2FA on that device (e.g., your home computer or other trusted devices), but you will need to use 2FA again if you login from a different device.
Beware: if you ever lose the app (for instance, you lose your phone, or delete the app, or do a factory reset of your phone), you may not be able to get to your 2FA codes when logging in from a new device. For that reason you should have in place some alternate method of authenticating yourself. **
Tip: when setting up the account and the provider generates the secret QR code, either take a picture of it or print it on paper. That way, you can setup the authenticator app again in the future without having to start all over. This is handy if you reset your phone, or if you switch to using a different phone, or if you want to setup the app on multiple phones--which can be convenient, say, if you and your spouse use the same Amazon account, for instance.
* The OTP apps are essentially the same, so you can use the Microsoft app for both services, the Google app for both services, or any of a number of third-party apps that do the same thing. I prefer
andOTP, which I use for all of my Microsoft and Google accts.
** Using each others' email addresses is one form of alternate method, provided (as you've discovered) you aren't locked out of both at the same time. Microsoft and Google can also give you a list of one-time use codes to print and stick in your wallet, for instance, in case all else fails. (I use that as my failsafe method if I should happen to lose the authenticator app.)
