Is an EF 16-35 mm full frame lens on a full frame camera equivalent to an EF-S 10-22mm lens taking into account my crop factor?
In some ways, yes. In other ways, no.
What we refer to as equivalence is only an approximation. You can't put a different lens on a crop sensor camera and get the same shot with the same field of view from the same shooting position with the same depth of field using the same ISO and the same shutter time as you can get with a full frame camera. The converse is also just as true. You can't put a 16-35mm lens on a FF camera and perfectly replicate the same shot you got with a 10-22mm lens on an APS-C camera. We can stop the FF lens down to match the DoF we got with the EF-S lens, but we then have to change the shutter time or ISO to get the same exposure brightness. To get a number of the variables to match, you must be willing to allow other variables to differ.
Having said that, to even begin to get close to the look you can get with a 16-35mm lens on a full frame camera, you must begin by using a lens with a similar field of view on your APS-C camera. This means a lens with a focal length of around 10-22mm. So in terms of field of view the EF-S 10-22mm f/3.5-4.5 on your APS-C EOS camera is basically the same as one of the the EF 16-35mm lenses used on a full frame camera.
To replicate the depth of field of a 16-35mm lens at f/2.8 on the FF camera, you must use an aperture of around f/1.8 on the 10-22mm lens + APS-C body. Unfortunately, there are no 10-22mm f/1.8 lenses available in the Canon EF mount. If there were any such lenses available they would likely be considerably more expensive than any version of an EF 16-35mm f/2.8. It is quite possible the price difference would be even greater than the price difference between your APS-C camera body and a Canon FF model such as the 6D.