Should I ignore her and do my own thing now?

Supposedly a week after we got out of school for college we broke up
She wants to keep talking the way we do
We even went hiking. We were acting as a couple. Bought condoms and had sex as well. But still she doesn't want to be called my girlfriend
That night I went on a date with someone and it went well
My ex text me and I haven't replied back and she even snapped me saying hello.
I'm done wasting my time. Is it rude to move on without telling her?
I still love her and she loves me and we broke up because her dad. I have to take him out on a date with coffee at the end of the month so I can start taking his daughter out. I'm still down for that but I don't want to talk to her because it's as if we're still together.
Should I ignore her and do my own thing now?
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