If a class interface doesn't change, but its implementation changes, (and accordingly the library file containing the class' functionality code) does the client code (driver app) need to be recompiled?
Generally speaking, no. If you don't break the ABI, then there is no need.
how would the executable file locate the new location of the edited method in the new library
The same way it did in the old library: the dynamic linker sorts it out for you as your executable boots up.