...I should be fine by then=...l may be fine by then?
Does"The doctor said it will take six weeks and I should be fine by then."(I got this sentence from the definition 4 of should in the Collins Cobuild Dictionary)" mean the same thing as "The doctor said it will take six weeks and I may be fine by then."?

4.[MODAL] You use should when you are saying that something is probably the case or will probably happen in the way you are describing. If you say that something should have happened by a particular time, you mean that it will probably have happened by that time.
"The doctor said it will take six weeks and I should be fine by then."

Question

...I should be fine by then=...l may be fine by then?
Does"The doctor said it will take six weeks and I should be fine by then."(I got this sentence from the definition 4 of should in the Collins Cobuild Dictionary)" mean the same thing as "The doctor said it will take six weeks and I may be fine by then."?

4.[MODAL] You use should when you are saying that something is probably the case or will probably happen in the way you are describing. If you say that something should have happened by a particular time, you mean that it will probably have happened by that time.
"The doctor said it will take six weeks and I should be fine by then."

Answer

In this case, "should..." means "will most probably...." "am/is expected to be...." For example, someone asked me "Will you be free to chat on Monday morning?" I replied, "Yes, that time should work for me" meaning, "That time will most probably work for me" or "I expect that time to work for me."

When a speaker uses the word "should" they are 99.999% sure that what they are saying "should happen" is most likely going to happen. However, this still leaves a little bit of wiggle room in case what should have happened doesn't happen - so that they can not be blamed for being wrong.

So.... in your example "The doctor said it will take six weeks and I should be fine by then." The doctor is 99.99999% sure that you will be fine after six weeks. But at the same time, he is leaving his statement open incase you are not fine after six weeks.

I hope that makes sense! If you need, I can try to explain it some other way! Let me know :)