Dear Mr. T:

You wrote:

1-We only don’t have to wash the dishes.

2-We don’t only have to wash the dishes.

3-We don’t have to only wash the dishes.

4-We just don’t have to wash the dishes.

5-We don’t just have to wash the dishes.

6-We don’t have to just wash the dishes.

Which of the above mean:

a-We don’t have to wash the dishes but we have to do everything else.

Which mean:

b-We can do other things as well. We do not need to limit ourselves to washing the dishes.

and which mean:

c-Not only do we have to wash the dishes, we have to do other things as well.

Again, this may vary slightly depending on context. In each case ask which word is the “only” modifying?

 1-We only don’t have to wash the dishes. This is awkward because it modifies a negative. It really serves little purpose here. Perhaps it suggests “a,” but there is but there is not enough context to say for sure.

2- We don’t only have to wash the dishes. This would normally be followed by another clause or sentence. The implication is certainly “c” because “only” modifies the imperative “have to.”

3-We don’t have to only wash the dishes. This also would normally be followed by another clause or sentence. Because “only” modifies “wash,” it might suggest that you have to do something else to to the dishes, i.e., We have to dry them and put them away. However, depending on the context, it could be seen as modifying the whole expression “wash the dishes,” so that it is suggesting that you not only have  to wash the dishes, you have to do other chores as well.

4-We just don’t have to wash the dishes. This depends on the emphasis. It could mean the same as #1, but the word “just” is a little stronger here so it could be spoken with emphasis. It does not suggest that we have to do anything else when spoken with emphasis.

5-We don’t just have to wash the dishes. Same as #2.

6-We don’t have to just wash the dishes. Same as #3