I think that it is a change of situation, but then I can't understand what the change is here. Please, explain to me
When 了 is placed at the end of a statement/sentence, it either challenges the validity of the statement or confirms it.
For your case, it indicates something contrary (to the statement "沒辦法") is true, or change is about to come.
- 不讓我送就以為我沒辦法了? The voice in the mind says, (辦法多着呢, 看這招!)
On the other hand, 了 in the sentence below confirms the statement before it:
- 我沒辦法了 - There is nothing I can do (now).
This single sentence does not give full context, so its possible that more clarity is needed. However I think that 了 here is showing that the events of the verb right before it are complete aka perfective aspect, rather than a change.
Since some of the context is missing, assumptions on anything not physically in the sentence are bolded below:
They did not allow me to send it, so I just believed (and was wrong) that there was nothing I could do.
Note that I have written this in past tense in English, as that seems most likely. However Chinese does not have tense so theoretically it could be in the present or future if that was the context of the sentence.
The allset wiki page below on uses of 了 may help you if you haven't seen them before. specifically the second link on the page for change of state with 了, the third link about to happen with 了, the fifth link already with just 了, and the ninth completion with 了. These cover all the vital basics of 了 as it is related to change of state or completion :)
不让我送就以为我没办法了？= 你不让我送，你以为我没别的办法了？= Don't you think I have no other ways of seeing you off?
The question is: what kind of function does 了 have here?
了, in general, has the meaning of "have done", "finished", and "completed". It can follow a verb to indicate the action is completed, or attached at the end of a sentence or clause to indicate the whole event is over. For example,
(1) 我吃了两个苹果。= I ate 2 apples.
(2) 家里没有苹果了，都吃完了。= There are no apples in our home. All apples are eaten up.
In the sentence of the question,
我没办法了。= I have no other ways to do it. = I got nothing after I thought about all of the possibilities.
了 means “it is over” and “I can do nothing else”.
The sentence (不让我送就以为我没办法了？) is difficult to understand due to the following issues.
(1) It contains two sentences and both omitted the subject 你。
你不让我送。= You do not want me to see you off.
你就以为我没办法了？= Therefore, you think I have no other way to do it.
(2) 以为 means to think. But in many cases, it implies that what you considered is not true or correct. For example,
不要以为你自己总是对的。= Do not believe that you are always correct.
了 can be "-ed" or "over". (have done something, for example: 我做了某事。)
in this sentence, just like: "you do not let me to do ..., and you think I have no idea for to do ..." this time, "the you thought the me have no idea", the 了 attached at the end of the sentence to indicate the whole event is over.