Welcome to the forum!
The tank should empty by gravity when the exit seacock is opened. It's possible that the pipe or seacock is blocked. If you leave the seacock open and motor around in choppy waters for a bit, it's possible that the blockage may clear itself. Alternatively, you could get the tank contents sucked out through the deck fitting, rinsing the tank several times while doing this, and then you'll have less effluent to cope with when you physically unblock it. This can be done with a flexible spiral drain clearing rod, from the outside of the exit seacock. If you're somewhere warm enough to get in the water, you could try this yourself. Otherwise, it's a nicer job if you get the boat lifted out first. Alternatively, if you're built like a gynaecologist, you could try putting a flexible drain rod down the pipe from inside the tank, getting access through the inspection hatch.
I wouldn't recommend trying to remove the outlet hose from the exit seacock - this will result in effluent getting in to inaccessible bits of your bilges and you'll never get rid of the residual smell.
To avoid this happening in future, make it a rule that absolutely nothing goes in to the toilet unless it's passed through a human first! No toilet paper, wet wipes, cotton balls, etc, these should all go into a nappy bag and be put out with the rubbish.
