Answer having wet grout around the drain does suggest that the weep holes at the waterproof membrane level might be plugged.
Shower floor drain weep holes.
Here s a short video showing what happens when weep holes in a shower pan are caulked shut the shower has a marble surround but it applies to tile coverings as well.
Weep holes provide a valuable service in draining away underfloor moisture that has seeped through grouting and corners.
Weep holes are a vital part of your shower drain assembly.
Doing so allows water to freely flow through the plumbing.
Another area that may need occasional cleaning are the weep holes that are located under the shower pan.
The weep holes are small holes that connect to the main drain allowing water that enters them to feed into the drainpipe.
Water below the tile that can t readily drain through the weep holes into the drain causes the tile assembly in that area to be saturated with moisture.
Line up the holes in the universal drain extender kit with the screw holes in the original shower drain.
If water penetrates through the tile grout and joints the weep holes help it to find its way into the drainage system.
Typically there are three such holes in the upper portion of the drain assembly spaced evenly around the drain.
It will then take longer for it to dry out.
Most home owners know that it s important to occasionally clean the shower drain.
Yes those are what we call weep holes they are necessary as you guessed to allow the water to drain out of the frame and into the shower.
When weep holes in the shower drain become blocked or clogged a disaster may be in the making.
I would suggest spraying a bleach solution into the holes from time to time to try to inhibit the mold growth in the framework.