To fix an overwatered aloe plant you will have to dry out the soil around the plant and also improve drainage of the soil.
When your aloe plant or any other plant is overwatered you should first stop watering and then repot the plant and ensure the new soil in the new pot has good drainage.
You can also use a stick and poke some deep holes into the plants soil to increase drainage and lead air down to the soil to speed up evaporation of the water.
If the pot the plant is in has no drainage holes you will need to repot the plant and use pots that have holes.
Remove the old soil and add fresh soil and inspect the roots of the plant and remove any mushy or discolored plant roots.
And use a well draining potting mix.
Allow the plant to air dry for a few hours, either in it's new pot or with the root ball of the plant exposed and you can also use a fan or move the plant to a location that has better airflow and if possible, gently loosen the soil in the pot to create air pockets.
A plant that is overwatered will have leaves that look yellow and the leaves of the overwatered plant will often be wilting and or drooping and the leaves will also often be soft and limp.
The overwatered plants soil will also feel wet and may even have a musty smell or mold or fungus may also appear on the surface of the overwatered plant.
Yellow or brown limp and droopy leaves is a sign of the plant being overwatered.
And if the leaves of the plant are crispy and dry then it's a sign of too little water.
Any wilting leaves that are combined with wet soil often mean that the plant has root rot that has set in and the roots can no longer absorb any water.