Your logic is flawed in one major way.... banks do not want to loan out money to businesses that are not good credit risks.... so, your thinking of lowering rates to lend more money is not logical to a banker... IOW, they are loaning out as much as they can with the credit ratings they like... and interest rates to lower credit rated people would mean higher rates for them....
So, you have deposits and nobody to lend that you want.... now, instead of getting some money for these deposits they are costing you... well, that is not going to happen.... so you have to pass your higher costs of deposits on to the customers with either higher fees or higher interest rates... it is not a stretch to think that higher interest rates are going to be in the mix...
As for just not taking deposits... the large banks can more easily shed deposits quickly as they just do not buy them... but you would be surprised how sticky retail deposits are.... most customers do not shop around for interest rates.... and for someone with maybe $10K to $50K a few BPs will not make them move their money...
So yes, I do believe that they would raise rates if they can.... now, a lot of loans are based on prime or LIBOR.... I just checked and LIBOR has been going up steadily since Oct.... so that kinda confirms they are charging higher rates.. prime looks like it just went up with the FF rate...
London InterBank Offered Rate (LIBOR) History