I do not agree that only the present perfect tense can be used with "in the last five weeks" or "over the last five weeks."
The question to ask in deciding whether to use the present perfect (example: "I have searched") versus the simple past ("I searched") is:
When did the activity occur in relation to the present moment?
If the activity is completely over (in other words: if the activity occurred in the past and the activity has completely finished), then use the simple past.
If the activity started in the past but is ongoing (in other words: if the activity occurred or was occurring in the past but has not necessarily completely finished--and it might still be occurring even right now), then use the present perfect.
The scenarios implied in the three sentences of the original question leave a bit of ambiguity (meaning that different interpretations are possible) as to whether the activity should be considered completely-finished-in-the-past versus occurring-in-the-past-but-continuing-until-(almost)-the-present-moment. For that reason, either the simple past or the present perfect could be used, depending on the speaker's/writer's perspective. If he or she wants to convey the idea that the activity is completely finished, then he would use the simple past; but if he wants to convey a sense that the activity has continued right to the present, then he would use the present perfect.
(Note that there are other possible verb tenses, but for the sake of simplicity, we are discussing only the simple past and the present perfect here.)
This general guideline applies regardless of whether the preposition being used is "in," "over," or "during."
The ambiguity I mentioned comes from the fact that the activity began in the recent past (just five weeks ago) and was continuing until only yesterday. On the one hand, since a decision was made yesterday to buy a computer (and, presumably, to stop looking at that point), it could be said that the activity completely concluded in the past, so the simple past is appropriate. On the other hand, the activity continued right until almost the present moment, so the present perfect could also be used, depending on the speaker's/writer's perspective.
By contrast, in the following situations, there is no ambiguity as to when the activity ended in relation to the present, so the tense that should be used is more obvious:
In the last five weeks of his life, Winston Churchill wrote only one more letter.
Over the last five weeks before I was born, my mother searched for a crib for me to be able to sleep in.
During the last five weeks of 1929, many banks failed.
In each of the three sentences above, the activities being discussed occurred well in the past and had completely concluded by the time the sentences were being spoken/written. (use the simple past)
In the last five minutes, I have been looking for my eyeglasses.
Over the last five minutes, I have been answering a question on the stackexchange website.
During the last five minutes, the wind has been howling.
In each of these three sentences above, the activities had started in the past but were still ongoing until the moment they were written; they continued until the present, from the writer's perspective. (use present perfect--does not matter if it is with "in," "over," or "during")
In the last five weeks, I looked for the best deal on computers, but I stopped looking when I found one I liked yesterday.
In the last five weeks, I have been looking for the best deal on computers. Yesterday, I bought one at a very low price, but I am still looking for a second one to give to my daughter.