what are the effects on gravity if the object is dropped underwater?
gravity's pull is the same, you just have a lot more drag underwater so it falls more slowly
so are there any equations for the free fall acceleration of an object underwater?
There is also a Bouyant force acting on a body underwater, this acts to reduce the "weight" of a body.
To model dynamics underwater you would need to take bouyancy and drag into account. There are different ways of modeling drag, usually it is a lossy term in the inital differential equations which is depentend on velocity.
m(d2x/dt2) = (rhoB-rhoA)(g*V) - 0.5(Cd)(rhoA)(A)(dx/dt)^2
Where m = body's mass, rhoA = fluid's density, rhoB = body's density, g = gravity (may be assumed to vary with height -> which is where the "effects of gravity" come in), x = displacement, t = time, Cd = drag coefficient, V = body volume, A = body area.
If the height changes are vast, you would also probably want to make rho, V and A a function of x as well.
Be a bit careful with the drag term in the equation - some quoted values of Cd use, for example, wetted area while others use cross-sectional area. Check the definitions before using it.
You can analytically solve that, write up your own program to solve it numerically or buy a math program which is capable of solving differential equations - MathCad comes to mind. Option 1 will help you only in the most ideal circumstances, while option 3 is a bit more general. Option 2 is the most general but most difficult to learn, and will solve even your nasty partial integro-differential equations with the most unusual boundary conditions, etc.
Edit: Note that the equation here is for a fully submerged body. For a partially submerged one, break the first RHS term to the body weight and a bouyant force, the latter as per Archimedes' principle.
It's no different than an object dropped in air. Normally, though, one neglects the buoyancy in air, since what we drop is so much more dense than air. But an example where it cannot be neglected is a helium balloon.
Separate names with a comma.