I CP'd my response from your 2nd Tri post: I wouldn't. Save the dates are for big events, like weddings, not baby showers. I definitely would think a FB event/text/phone call would be extremely tacky.
Registries are pretty common and easy to find. If someone is that concerned about shopping sales for your baby shower, they will know what to do. Most people don't shop registries anyway.
Shower invitations should be sent 3-4 weeks in advance, but that's something that your hostess should handle anyway.
ETA: adding registry info to a save the date or texting/calling/FBing people with your registry information is going to come off as extremely gift grabby, even though you are doing it under the guise of saving them money.