If I pay “CASH” will my offer trump the other offers?

Yes, but only if all the other terms of your contract are equal or comparable to other offers.

While all sellers like cash offers they will not take less money just because it is cash, unless they believe that the property may have a problem with an adequate appraisal (cash transactions do not require an appraisal) or that due to the property disrepair, a lender may not want to finance the property.