a 'me too' here.
SMS does not give two shits about whether there is a WiFi or Data signal present. I always have "the internet" (data & wifi) turned off when my screen is locked.
SMS beeps me just fine and I have messages to read and can send a reply. All without activating "the internet". It is a part of the basic '1x' type phone service. Has nothing to do with any of the "smart features" of our newer phones.
If you have txt msg problems, you have carrier issues, or a phone that is not properly configured.
We can't see your phone, so you have to use our answers to figure out what is going on.
EDIT: Automatically turning the Internet ON and OFF.
I think this might be of benefit. This is a direct copy of a post "somewhere" in this S5 forum.
I installed MacroDroid and set up a macro that senses Screen Lock.
* MacroDroid - Device Automation - Android Apps on Google Play
Labeled:Ringer Vol Max
T: Screen Off
A: Set Volume Max
I also changed the ringer tone to "Old Telephone" which sounds just like the old black bakelite phones of the 30s, 40s and 50s. Very distinctive and I know it is my phone.
With the macro set up, it is always at Max Volume. However, you can set the volume level to your preference. I just moved the slider to Max.
I have other macros set.
Labeled Internet Off:
T: Screen Off
A: Data off, disable Wifi
Labeled Internet On:
T: Screen Unlocked
A: Data On, enable Wifi
MacroDroid has a huge library of transition sensors, and controls. You just have to play with it. Once you do, you will come to depend on it a lot.
For instance, I am now on day 3 since mine was charged. The battery is 65% while it is displaying my MacroDroid macros.
Turning the internet off while the phone is not in use has a tremendous effect at stopping wasted battery power.
It only takes about 2 seconds for Data ON to become effective, another 3 seconds and Blue Mail downloads all of my emails. For me, that is very inconsequential delay times to have a battery that is always full of energy to use "when I need it."