It's possible the water is getting around your shower, possibly where the tile meets the tub or pan (if you have one), or behind the fixtures, or even around the door (if any).
I would check all of these and recaulk where appropriate before going in for wholesale destruction. Try splashing water by hand/bucket on areas you might suspect and see if that leads to leaks.
Or maybe you even have leaky plumbing in the wall. Do you have access to see in there?
For any leak, you have to find the source, and also note that leaks travel along surfaces so sometimes where the water is seen is not directly under the leak. Often this will involve cutting up walls and causing more water damage.
You need to find out if the leak is coming from a supply line or waste line, and also when the leak occurs.
First thing - when there is no water running and no water in the tub, is there a leak? If so the leak is before a fixture in a supply pipe.
Now plug the tub and fill it up. While the water is running (but not draining), is there water leaking? If so, the leak is at or beyond the shower valve and possibly in the shower itself. You mention that the leak is pretty big, so if present at this point, stop filling the tub and watch to see if the volume of the leak subsides. If so it could be a leak in one of the shower valve fittings or the diverter stem, or it could be water splashing up and getting behind the tile. In this case your best bet is to open the wall behind the shower (if possible) and look for leaks.
If the water is still running with the tub full but off then the leak is in the drain pan.
If not, pull the plug and check again for leaks. If you see water now, the leak is in a waste pipe. This is the scenario where you can end up cutting lots of drywall to find the source. Start at the tub and go from there, follow the pipe.
If the drywall/plaster is already water damaged, you may as well rip it out before it grows mold - you are going to have to replace it anyways.
Best Answer
Vacuum it out, preferably with a Wet-Dry or Shop Vacuum that has a thin nozzle attachment applied parallel with the seam or a floor sealing Water Pickup attachment applied perpendicular to the seam. To assist this operation you can tape the seam so the vacuum's seal is concentrated. A bag-less vacuum can work just as well, but be careful not to fill it more than 1/3rd & damage the vacuum or electrocute yourself.
If those don't get most of it or neither are available to you then a thin towel or cloth, like bed sheets or clothing can be stuffed into the seam with a putty or butter knife to soak up & wring out until dry or mostly dry. Avoid paper towels as they can easily rip & are a pain to straighten back out repeatedly.
Then, let it air dry. If you need to use the shower then just duct tape the seam to keep water out for one. But for another, to see if the water is coming from somewhere else & the seam is filling up again. If it's unchanged then keep taping & un-taping until it's bone dry.
Finally, actually fill-fill the seam with caulk & force the caulk in with your finger. Don't bother trying to get it perfect the first time, instead come back in a couple of hours & do a second application of caulk to get the caulk flush to both the floor & wall. Then, don't use the shower for at least 6-hours. If you can't wait that long or longer then tape OVER, as in bridge, but don't tape ON the caulk.