How do I search the collection for a weakness?
Generally, yes. However there are multiple effects going on here:

1) You are being asked to search your collection (meaning your card pool) for a random basic weakness.

2) Then that card is being drawn. Since basic weakness cards are (typically) revealed upon being drawn, it would be played. Depending on the basic weakness drawn, it may either be placed in your threat area or your discard.

There might be effects in the future that operate differently (such as search your collection and add a basic weakness to your deck, which would be shuffled), but I believe in the coreset the above guidelines are the only ones applicable.