I recently switched from an 8" deep snare to a 5.5", and it takes a little time and a little experimentation to find a sound that feels comfortable.
However, there is a certain quality of sound that is unique to specific depths, and it is a little unfair to the drum to expect one thing out of another. I feel like the best snare sound is achieved by trying to maximize a drum's particular sound, not force it into sounding like something else. It may not be the exact sound you're wanting to hear, but at least it should be a good sound.
That said, I think most drums are capable of a wider range of sounds than they're given credit for, so tweaking tuning, heads, and wires is always a great pace to start.
It seems deeper is the main concern, so let's start there. The most obvious thing to try is simply tune the batter head to a lower note. The trick here is to control any higher overtones you're likely to get from the lower tuning. A piece of Moongel might do the trick, or a head with some built-in muffling. I'm using an Aquarian Studio-X batter, and it gives me a great combination of shell tone with slightly-controlled overtones.
I don't like overly thick or dry heads on shallow drums, since all I hear is slap from the batter head and buzz from the snare wires and not enough tone from the shell to fill out the sound. It's what happens in-between the heads that gives you the feeling of depth you're looking for and both the HD Dry and the EC2 kill too much shell tone for my liking.
As for dryer, I'm not sure if we're talking about cutting some high-pitched shell-ringing or if there is too much snare wire sound. If it is snare wires, just get a set of wires with fewer wires or remove some wires from the existing set.
If it is high-pitched overtones I still think tuning and some muffling is your best bet. I like starting with Moongel just because you can totally control how much muffling you get. The trouble I found with thicker and more muffled heads is you can't adjust the level of control at all. The HD Dry especially gives one sound no matter what tuning you're at. For me it killed the drum completely and it sounded very one-dimensional.