New owner got stuck in mud in main channel

"An obvious learning lesson is to use the depth finder"

Before you put to much faith in it, find out what it's really telling you. It may or may not have an offset put in it. So it could be reading true depth, keel depth or anything in between.

Personally I prefer a fish finder type display over a pure digital. Fish finders at least give you a picture of what is happening. Where digitals could be reading something other than the bottom. Many times I've watched the digital display go out on a fish finder while the graphic one keeps on reading the bottom just fine.

Did the engine stall as you were trying to back off? mOr after you got into deeper water?

Seajewel,
One thing about a depth finder it tells you about the depth you just passed, not what is coming up. Remember that transducer is probably 2/3 or more back from your bow. Check the depth finder about as often as you check your rear view mirrors in the car (every few seconds). In a shallow area one eye doesn't leave it. Watch for the bottom starting to come up (7,6,5,4,neutral).
While you're checking the props hold a straight-edge from your rudder shaft out to the top op the prop blade. Keeping the straight-edge steady spin the prop making sure all blades meet it the same. Often a slight bend can't be seen with the naked eye, especially under water. This method will serve you well. Also check the struts. If as you turn the prop you feel a rubbing that would most likely be your cutlass bearing.
The fact that the motor died several times indicates you may have ingested sand while trying to extricate yourself. You may need to back flush you motor and change your thermostat a couple of times.
About that training captain...what lesson was he trying to teach by turning off your depth finder? Find a 100GT captain on at least his 3rd issue (more than 10 years professional experience). Along with docking and the basics spend a day with him going to places you may have thought of going or places he thinks you should watch out for.
As for the E-80, it's a good unit. Try using a duel-screen (chart and radar and YES use the radar even when it's perfectly clear. That's how you get used to recognizing what you're seeing). The set up a data box for depth as was previously suggested. Move the range in and out. As you come in you will get more detail. On a day that you're just hanging around sit down and play with it, play with it, play with it. Put in routes to everyplace you can envision yourself going. That's the only way to make it second nature. Also, don't forget to set up the lolipop on your radar. If you get caught in fog you'll want your radar as big as possible. If you put in your route on the chartplotter the current leg will be on your radar so you won't need the plotter on except when you get into tighter quarters.
I think you just found out why they say BOAT stands for Break Out Another Thousand. Hopefully not to many this time.
BTW, watch for the change of texture and color to the water. That indicates a change in the bottom. Also, when in doubt proceed at dead slow or follow someone bigger than you.
Good Luck.

"Right below both engines in the ER compartments towards the interior of the boat (medial) I see some water mixed with oil. Any clues??"

Hard to say. May be something, may be nothing. Check the fluid levels in your transmissions. And look for signs of leakage around your tranny seals.

More people tend to damage something trying to get off the bottom than going aground.

I would not worry too much about this. From everything you've said it appears that you only wacked one prop. You also said you had an oil change recently. Odds are a few drops spilled into the bilge at that time. There will almost always be some water in the bilge from the shaft logs even if they are dripless (especially if they are new) or water from rain, washing or through the vents. Let a drop of oil mix with a few drops of water and bingo. Taste the water to see if it it salt or fresh. That will give you a hint at where to start looking for intrusion. Also, clean up that bilge. A new boat is easy to keep clean. That way if you do find a drop of oil just look up from it and you'll instantly know if you have a problem..

Sorry Pascal, but I have to disagree with you strongly about this one. I've met too many panicked boaters (with years of experience) stuck in the fog. The GPS can be followed by anyone who can read a map and follow an icon across a screen. They're the greatest thing since sliced bread, but when that fog moves in and you stop the boat you'll be spinning in circles. Also, remember that some units have up to a 6 second lag time. That radar will be the best friend in the world IF you're used to using it. That is NOT the time for learning though. As you head for your home set the radar @ 3 mi. When you reach the 1.5 mi. line range in. Then .75, .50, .25, .125. and drive between the jetty images. Watch the radar as you do this with only an occasional glance up (the opposite of what you want to do). You'll be hooked.
Props on e-bay. Although it's a good idea to carry spare props if you travel much (It could save you a 3 day layup waiting for delivery, but I've only changed 1 prop in over 20 years.) I would NOT buy these. Propping is a art & a science. Those don't match. Those will certainly raise your RPM's, but I have no idea of how they will effect your load, torque, speed, etc. If you want to get different results from your boat (faster out of the hole for instance) go to a reputable prop shop and let them figure out what you need. At this point you shouldn't need a new prop unless you broke a blade bad. In most cases though they'll add metal and bend it right. It won't be cheap, but it won't be 3K.
BTW, you may even want to skip the diver and just get a haul. You know that one prop is dinged so you'll be hauling to have it changed anyway, and it's much easier to check on land anyway.