I don't remember if Namecheap does, but most registrars I've used give an option when requesting a transfer as follows:
1. Leave the whois as is (your existing info).
2. Use the receiving persons info saved with the new registrar (Ideal).
3. Enter desired info.
If you use option #1, the whois (your existing info) will not change until the new owner manually changes it.
If you have existing whois privacy on a name with Namecheap, you can assign it to another name if it hasn't expired. However, I learned the hard way that you must disable the privacy for that name before requesting the transfer or you will lose the remaining time.
In addition, the saved personel info that is shown on whois (without the privacy) might be publicly displayed between the the time of the transfer request and it's completion when you disable the privacy.