Hi,
yeah, a cross bike would be fine. Touring bikes also accomodate larger tires, they'd work too. You may need a special rack that lets you get the panniers further back than a typical rack would allow.
Don't skimp on lighting, if you need it. Tires are a tough call. People have very different ideas on tires. I classify them by weight, 200 gr tires, 300 gr tires, 600 gr tires. Heavier usually means a lot fewer flats, and more miles, but of course... slower.
I like a 300 gr tire.