Is this what a leaking toilet looks like?
There are numerous explanations for possible causes of moist grout near your toilet base, one of them being a compromised wax ring.


washing/cleaning of the outside of the toilet can allow water to run down and moisten the area at the base
a compromised seal between the tank and bowl can allow unseen water to run down and moisten the area at the base
poor aim by male household members will cause urine to run down and moisten the area at the base, this would cause a urine smell as well (this is a thing, I had 3 boys in the house)
a compromised wax ring could cause moisture at the base of the toilet


Some modern wax rings have an integral plastic funnel that allows gravity to direct water/waste into the drain line, which helps prevent leaks (under normal circumstances) even if the wax is compromised. I would recommend that you carefully examine the unit for other signs and if you find none go ahead and pull the toilet. Get a friend to help, they are unwieldy, and use a ring like I mentioned above when you replace it.