My airway had multiple problems that had to be addressed.
1. Block sinus drainage pathways.  I at least sought treatment before
any more serious complications.
2. Moderate left side obstruction from previously broken nose.
3. Enlarged turbinates in the nose from chronic allergy problems.
4. Significantly enlarged and growing tonsils as an adult (not normal).

While I cannot recommend surgery to anyone unless there is simply no
other option and you seek a couple of different opinions to look at the
benefit versus the risk, most of my problems required surgery.

1. Septoplasty to straighten out the nose, while the ENT was there he
reduced the turbinates with Somnoplasty, and opened up the sinus
drainage channels.
2. About 8 months later the tonsils had to come out, after delaying
this almost three years.  As it turned out the tonsils had started to
grow into my cheek, lower gums, soft palate, and one side was nearing
the voice box.  This was a significant and painful surgery recovery.
The plus side was due to the tonsils into the soft palate near the
uvula, a soft palate resection was required to get everything back in
place.  4 weeks of serious pain and healing, followed by a couple of
months of final healing.
3. Very careful control of allergy symptoms.  Careful choice of bedding
and being proactive with prescription antihistamines and occasional use
of prednisone during the worst allergy times, grass pollen and corn
pollen, now prevent the sinus airway problems from allergies.
4. Control of minor acid reflux, especially at night due to hiatel
hernia.  No eating food within two hours of bedtime.  Pecid AC at
night.  Raise entire head of bed a couple of inches.

I would still explore if you notice you breathing through your mouth.
Even during the day, if you have any problems through the nose, you
will breath through the mouth and might not notice it right away.  At
night it is a simple matter to video tape yourself with a modern video
camera in IR mode, then you will know if you breath through your mouth
at night.

There is available a simple chin strap to help hold the mouth closed at
night.  There is also a mouth guard available through many dentists
that will help hold the mouth and jaw in position.  The dentists guard
is useful for TMJ, grinding teeth, and there is also a mouth appliance
that will help hold the jaw forward and tongue forward to assist with a
reduction in snoring.

You can do a search for burping/belching and swallowing air for many
possible causes.  Even minor anxiety and stress can cause some people
to unconsiously swallow air, as can constant breathing through the
mouth, chewing gum, and too many other causes to list here.  Do a
search.