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Hi, I'm Nathan.Sorry to hear that you're having trouble, but I'll do my best to help you. Please give me a few moments to review your details here.
Unfortunately that is simply not possible.
Your vtech does have an answering machine, but this is just a simple single line phone system. It is not possible for your vtech or any other phone for that matter to have that type of function with just a single line. Doing that would require a multiline phone that would roll incoming calls over to a 2nd line.
In your situation, if you don't answer call waiting, they'd just get a busy signal. The only possible way they could leave a message is on the ATT voicemail. You could switch back to using the voicemail and turn off the vtech answer system instead if you want that functionality.