We have pay-as-you-go regular cell phones (no data plan or contract, no smartphones or tablets or land line). Diablo will not allow us to use our phone numbers when we try to sign up to be able to use the RMAH. Is there anything we can do to legitimately participate in that part of the game without having to change phones and carriers?
There are two ways to use the RMAH, once via PayPal (which needs SMS Authentication) or via Battle.net Balance which only works if you have an authenticator attached to your account. So since you can't use your mobile to use SMS authentication you are left with Battle.net balance and an authenticator attached to your account. Here you have 2 options:
I just want to offer a suggestion to use winauth. It's a very useful application. I can also attest that it works fine for people without a smartphone. I passed this website to my friend who like you has a dumb phone and has no problem using it for his account.
The other thing that I find useful.I have the authenticator on my android smart phone, but sometimes I'll leave my phone upstairs in my house or charging somewhere in another room. Once I changed the serial and authenticator code to match my phone. I now can always pull it up on my computer if needed. Also includes a handy send to clipboard function if you are logging in via a browser. There are also options to encode the serial and put a password on the application if you would like as well. Overall a very useful program.
The other useful thing I want to comment on as well is you can write down your serial code for your authenticator and if you are ever in a spot where you don't have you authenticator on you, you can download the application and get your key as long as you have that serial code (saved somewhere securely I hope).