Nvidia core drivers are all the same, mostly directX etc. The biggest differences come in the extras, mainly game optimizations, and it's these that should be replaced as what's optimized for 1 card will be different for a more powerful card. This is why when installing the new gpu drivers, it should be custom, not express. On a page right after that choice is an option for clean install. What express does is only replace older versions of drivers, and add any new optimizations, it doesn't change or overwrite anything it sees as already installed. This can lead to issues unless you are just updating. Custom allows clean install, which removes all the drivers and optimizations, installing fresh copies.
No, changing your hardware doesn't change the drivers. What it can do is revert to default settings as the new gpu isn't fully recognized as such, relying on core drivers. However, the 760 is pre-maxwell and the 1060 is pascal, so anything other than the core drivers is going to be in conflict as power requirements, instruction sets, optimizations etc for the 2 cards are different. This includes the gpu bios, which is legacy on the 760 and Eufi on the 1060 which can and frequently does cause many conflicts.
Best bet is use DDU or at a minimum custom/clean install of the correct drivers for your card.
Psu should in no way be an issue since recommended psu is 500w for a gtx760 and 450w for a 1060 unless you have some really oddball cpu like the FX9590, in which case you can add 400w to both those requirements.