Attorneys must electronically file all documents in cases with the Texas Supreme Court. All persons filing electronically must also file printed paper versions of the file-stamped e-filed document within three business days: one for the record of an original proceeding, two of petitions, responses, replies and amicus briefs, four of any brief on the merits. SCOTX has some helpful resources on its website for new to electronic document production and practice.
Now you have to pay an e-services provider to file any paper. That fee is larger than mailing fees, but probably saves money overall and in the long run because of reduced costs related to producing more paper copies, but in the short run you have to pay the provider and prep up some copies for mailing, and mail them. Large corporations can probably deal with the electronicnization of document production OK, but my family law and probate clients can't deal with it so well (My criminal law clients don't have business with the Austin Supremes.).
Some good news is that if your back is absolutely against the wall on a filing deadline, you can now file late at night the night the document is due (But don't do it, always leave time for someone's computer to crash, for you or a courier to have a flat tire or other car trouble, etc.).