Dear N T:

You asked:

1-You do not have to go to only one of these cities.

2-You do not have to only go to one of these cities.

3-You do not have to go to one of these cities only.

4-You do not only have to go to one of these cities.

Which of the above mean:

a-You can go to more than one of these cities.

and which mean:

b-You have to go to more than one city. Or maybe, you have to do something other than going to one of these cities?

The words “just” and “only” are notorious for being difficult to manage without context. However, what is important is that the “only” normally modifies the word that follows it. (Obviously, #3 is different.)

Therefore, #1 means that you must go to more than one city (i.e., not only one).

#2 means that you must do more than just go to one of the cities (i.e., not only go, but do something else).

#3 means that your choice of cities is not limited to these cities (i.e., not only these cities but others, too).

#4 means that you must go to one of the cities and do something else (i.e., you not only have to go, but do something else). Because “have to” is modified by the “only,” this means the only is modifying something that is imperative.

Having said all that, occasionally in everyday speech people will put an “only” where it technically does not belong, but it is understood because of the context.