I would not add a new flooring down, you will have door closing problems, wall problems with molding, and in general, your floor would be 3/4 inch higher if you use that width of new flooring.
I would tackle the areas that are mushy. Apparently your old flooring is that glued particle board material. It does not like water and turns to oatmeal with the introduction to water.
Cut out the mushy areas to a perfect square or rectangle and consider the joists that you need to nail to, or what ever a mobile home has. then add the new flooring in that square and make sure it is the same thickness as before. Seal the cracks with a good 100% silicon caulk.
Your mushy flooring should be in areas around water use, like comodes, sinks, dishwashers, cloths washer, and the such.
You can do it. No big deal.
a few tools that you will need include:
A reciprocal saw. a screw gun or drill. caulking gun. a pry bar. knee pads.
Ps, be sure to go under the home and make sure that all cutting will not cut any pipes, electical lines, or other important stuff.