Ummmmm ..... no way to know what your rims can fit because you never mentioned rims .... but if your rims were designed for 15-mm tires, which as far as I know are not even produced or used (except in C&V-land) then your wheels are probably old and tired and need replacing.
If you mean your wheels have 15-mm internal width, you should have no trouble fitting 28s, assuming the 28s clear the frame. I would not want to do much off-road with 28s .... but "offroad" is not a definition it is a description. Rock-hard baked mud, no problem. Sand, loose gravel, pea gravel, single-track? Game trails? Pastureland?
I suggest tires built to ride the terrain you plan to ride. No idea what that is.
I have done supported coast-to-coast, on pavement, on 23s ...... but all the gear was on a vehicle and all the roads were paved. And a few times i wished for wider, softer rubber .... chip-seal is brutal. On the same ride I used a second bike with 32s and it was much better. I also did a south-north tour, fully loaded, on 32s and again exclusively on pavement. No problem.