We reached the maximum number of searches in our Media Injection account. Is it possible to delete full or quick searches without losing data?

In total, you're allowed to have 3 active full searches and 10 active quick searches. Besides the active searches, you're free to add more searches. However, you can't add a new search if you're on the maximum number of active searches. Disable a search that's not longer relevant. When you disable a search, statistics will be saved. Because no it's no longer an active search, you're able to add a new search.