Please install the latest available version on the old system. Create a backup of the settings ("Configuration"->"Export settings..."). Now install the trial version on the new system and import the settings ("Configuration"->"Import settings...").
As soon as you like to deactivate the old server, you may uninstall SmartPOP2Exchange and install the full version on the new system. The settings are not changed by installing the full version.
When using online Exchange Server/Office 365 you need enter the SMTP server name of your online Exchange Server.
This is not "smtp.office365.com".
For a domain like "myDomain.com" the server name is usually composed like "myDomain-com.mail.protection.outlook.com".
To get the correct SMTP server address you need to do a lookup for the registered MX server.
The SMTP server you get for your domain MX lookup has to be used as SMTP server in the SMTP setting of your SmartPOP2Exchange account.
Keep using port 25 and do not use authentication.
Additionally you should add the externally visible IP address of the server running SmartPOP2Exchange as accepted IP in the connection filter settings of your online Exchange Server.
Since Exchange Server 2007 the exchange checks if the user of the authentication details used for the SMTP connection has the privileges to send mails using the address of the FROM field in the message.
With Exchange Server 2007 and newer you should NOT use authentication for the SMTP account settings in SmartPOP2Exchange.