Snowbear:
The deeper you dive, the more your body has to work against the higher ambient pressure to expand your chest and draw a breath.
That is not the case. The ambient pressure outside is higher, the pressure inside your lungs also. They compensate (in fact if they did not your lungs would collapse).
So you breath air at higher depth at higher pressure. Air at higher pressure is more dense (it contains more air particles per unit volume). This makes the air more sluggish: you have to breath harder to fill your lungs.
If you ever end up in a deco chamber try to whistle at high pressure. You won't get any sound out of your mouth because of the denser air. BTW: I really hope you never, never have the chance to do this experiment...
Firediver:
Your lungs become compressed at depth and your body is probably trying to compensate for the lesser space
Your lungs do not get compressed more or less than when you breathe at sea level. Your lungs only get compressed when you hold your breath and go down. When you normally breath, your lungs do not get compressed.
Original question:
So the air being dense makes you have to work harder. This might perhaps (just guessing) give you a feeling of shortage of breath and might stimulate you to breath deeper.
Some (most?) regulators blow air into your mouth, after an initial effort to start breathing (you have to overcome the resistance, then air starts to flow on itself). It is interesting to know if your 'problem' occurs using such a regulator. Usually, when I switch my regulator for my snorkel, this feels awkard, because I have to suck in air myself instead of that it is blown in my mouth.