Redying T-shirts

You should purchase professional dye from an art store. You'll need soda ash and salt. There should be directions on the package on how to do a cold dye bath. This would probably be the best way.

How old are these shirts? If they are so old they're fading, I'm not sure dying them is even worth it. T-shirts are meant to be disposable.

You generally can dye cotton or cotton/poly shirts without affecting the ink printing.

I would recommend you do NOT use RIT dye but instead purchase a higher quality dye from
http://www.jacquardproducts.com/products/dyes/idye/

Hope that helps