Oh, Man. That looks so familiar. My truck looks like that three or four times per hunting season.
After a lifetime of doing things like that, let me give you a few tips.
First, don't even check. Just go ahead and change the oil in the front and rear ends, transfer case and transmission. Shaft seals are primarily designed to keep things in, not out. So even if your vent hoses are up high, that much hydrostatic pressure will force water into places it should not be.
Be sure and check your engine oil level. If it is even slightly too high, or suspicious in any way, change it too.
If your rig has a clutch-type cooling fan, earmark some money for a replacement down the road. You'll need it.
Before you drive it any distance, clean the front end up good, and then get under and start doing a meticulous inspection.
Pay close attention to the spring perches, shackles and bolts, and where the shackle mounts attach to the frame.
I would not be surprised to find bent shackle bolts, sheared kingbolts, and cracks in the frame where the springs attach.
I know you folk spend lots of time and money making your suspension bulletproof, but trust me, when you start talking about serious winching, your suspension is the weakest link.
That is why, when you are seriously stuck in deep mud, if at all possible, you MUST have the engine running, the transmission in gear, and the transfer case in low so the wheels are turning all the time you are being pulled out. Those tires have to dig a trench as they go, otherwise you run the risk of leaving your front end in the hole.
I realize that you could not do that, which brings me to the next part.
When you first drive off into something like that, its ok to try briefly to get out under power. But don't be a fool about it. If you are digging deeper without moving, stop.
Then, wait a few seconds to make sure the truck is stable. If it is sinking and is past the bottom of the doors, (been there) roll the windows down and get the heck out.
If its not going down, and the mud level is door blocking level or above, pop the hood release if you have one, go out the window and slog around to the front.
Remove whatever articles of clothing you don't want to ruin first, and cinch your shoe laces down tight to keep them on your feet.
Check to see if the mud is blocking the radiator, and while clearing it, dig down to the winch if you have one.
Then, open the hood, and check the fan.
If its hitting the mud, and is not a clutch fan, you have a problem. If it is a clutch fan, you can run the engine for short times, but you will have to buy a new clutch sooner or later.
If not a clutch fan, and you have some tools, see if you can remove the fan and put the bolts or nuts back on to let the water pump spin.
Of failing that, see if you can dig the mud out deep enough to free the fan.
The important thing is to be able to run the engine while getting winched out so as to not damage the suspension.
If the cooling system is compromised, you will have to pull out in stages, letting the engine cool off in between.
If you are going forward, stop and check for mud buildup in front from time to time. I have seen radiators pushed into fans by mud.
If pulling to a tree, try to hook the cable as high as tree strength allows to let the pull help raise the truck.
There are several things more, but this is getting long, and I am due for a nap, turkey season being still in effect, so I'll stop here.
J.