If the water isn't actually going DOWN that white tube and continuing to run (i.e. overflowing into the white overflow riser or tube), then the water level setting is probably okay.
If the black tube is above the top of that white tube, and not shoved down the white tube, then that's okay. If the black tube is shoved into the white tube rather than being on a clip above it, that's a common cause of a teensy bit of water running out the fill valve. It creates a siphon and pulls water from the tank through the fill valve; hence the little clip to hold it above the top of the overflow riser.
I should say that it's kind of normal for a few seconds after the bowl has refilled and the tank shut off to hear a little water dribbling out of the bowl. Not unusual at all. But if it's dribbling out of the tank into the bowl more than a minute after the water has shut off, and is enough to cause the toilet to turn on on its own and fill the tank, then it's most likely that you need a new flapper.
The easiest way to tell is this. Turn off the water to the toilet at that little knob on the wall. Mark the water level with a pencil on the tank. Won't hurt it. Wait an hour or two or three. Come back to the toilet. Has the water level dropped a material amount? If so, then you probably need a new flapper. Go get yourself the Korky 3060BP adjustable 3" flapper at Home Depot or Lowe's and follow the instructions. Or order a Toto THU347S flapper online (NY Replacement Parts has it online for like $8 plus shipping. That one you won't have to adjust. (The THU347S flapper is the right one for your Eco-Drake.)
If the water level DOESN'T drop when you leave the tank with the water off, and you hear a little flow from the fill valve, perhaps enough that it's starting to pour over the top of the overflow riser, then it's a fill valve issue. Easy enough to repair that. Even total amateurs can do it. The proper fill valve can be purchased at Lowe's (Korky 528MP or 528MPK with the silver cap; the white cap regular 528 one won't fill your bowl enough) for like $11. If you get to that stage, then come back and we will walk you through how to install it. It will take you at most 15 minutes, basically with no tools. My Mom could do it. If you currently have the Korky fill valve in your toilet (the so-called Type B Valve in this piece: http://www.bay*******************/ima...ns-rev06-1.pdf), then you can repair it by changing a little cap inside it which is available for like $4 at Home Depot. That's an even easier repair.
Anyway, let us know how that first test goes, with the line and turning off the water, and we'll take it from there.
PS So you can sound like a real pro, here's the actual model number for your toilet: the number you gave us (the number in the tank) is the model number of the tank. Toto model numbers for a two piece toilet are a mashup of the tank number and the bowl number.
So, if you have an elongated Eco-Drake, then you have tank ST743E and bowl C744E (or EG if you have Sanagloss as well), for a toilet that is CST744E or EG.
If you have a round Eco-Drake, then you still have tank ST743E, but you have bowl C743E, for a toilet number that is CST743E.
There are some other permutations of the numbers based on options like which side the handle is, but those are the basics.