I don't think it necessary, in general, to print of the content of all e-mails; you just need to show that you have been in frequest communication, so the e-mail headers should be adequate.
I am sending SMS to my fiancee by computer - it is cheaper than sending from my phone and it also keeps a written record which will be submitted with the Visa application. But, again, I will not send contents of the SMS - just a record of when the SMS was sent.
You may want to enclose a few e-mails in full over the whole period you have been in a relationship - just to prove it is a 'romantic relationship' rather than just a business one.
After all, in the days before e-mail and SMS you would not have been required to send a recording of phone conversations to prove a romantic relationship!
