What document do you need to search another employer? You don’t need any document to search for employer. Yes, the new employer can ask for proof of approval when they plan to file H-1 for you. At that time, you can talk to their attorney and provide them the receipt number which shows your online status as approved. They can then file the new petition for you using the receipt number. This petition goes to USCIS who already know about your previous approval as that also came from USCIS. So it is ok not to have the actual approval notice in hand (although some employers prefer to see that before spending dollars on your filing).
The current employer can write to USCIS to withdraw your approved petition. However, that doesn’t changes the fact that you have already been counted in the cap. So another employer can still file for cap-exempt petition for you.