Step One: Attempt to secure you computer, if an attacker has compromised your computer they could be using key-logging software to capture your passwords or remotely control your session:
- Run two virus and malware scans, from well know, trusted vendors.
- Remove any unused software.
- Remove any unused extensions or plugins from your browser.
Step Two: Ensure your gmail account is configured securely. Most password reset functionality trusts only you can access your email account. If someone has access to your email account they will be able to access most of your accounts.
- Change the password
- Set up two factor authentication on your account.
- Check which application have what permissions to your google account via https://security.google.com/settings/security/permissions?pli=1
Step Three: If you have any Android devices associated with this gmail account, review which app's you have installed and what permissions these have. Remove any app that your don't use or has more permissions then it requires.
Step Four: Check your security settings in FlipKey, I don't know FlipKey so this are general advice:
- If you can change the login provider to one of the big names do so. (logon with [Google, Facebook, Yahoo, etc].
- Otherwise change your password, and if FlipKey provides two factor authentication enable this.
- Have a look through your account page and see what other security features FlipKey provides.
If after this you still have an issue I would follow the advice of @AntoniusBloch and use a separate computer to change your passwords again and wipe your computer, and reinstall.