On my last phone (GS2) and my current phone (GS4), it has worked that the 3G/4G icon goes away when you connect to WiFi. I don't think that means you disconnect from your cellular data signal, only that the data transfer service has been handed over to the WiFi radio. When you turn off the WiFi radio (or when you're out of range of one the saved signals), it will hand those duties back over to the data radio. From what I know, your cell signal is still used to send SMS and MMS messages when connected to WiFi. And in my experience, that has been true. When I've had good WiFi access, but in an area with a poor data signal, text messages tend to have a problem being sent. If you're having similar issues, my guess is you're in an area with a poor or busy data signal and you'd have these issues whether you're connected to WiFi or not. That's been my experience any way.
By the way, if you lose your cell data signal, you'll get a circle with a line through it in place of the bars. Since you have bars, you're connected to your cellular data.