Can I use a different mailing address from my residential address?

You can use any address that you wish on your tax return.  If you have a preferred mailing address then you can use it.

Keep in mind that if the IRS or your state department of revenue needed to contact you about your return, they would use the address on the tax return. 

Turbo tax will not ask me for information to enter the amount I pay for assisted living at another address.

Please clarify your questions.  Why do you need to enter another address?  Please ask your question with detail.