I have not heard that constipation is a sign of gall bladder disease. If anything, you usually get diarrhea. However some people vomit their bile out rather than poop it out, so those people might have regular or even constipated bowel movements. I have always been a bit on the constipated side, so I could be constipated between gall bladder attacks, but when I got sick and started to hurt the diarrhea always came on.
It is quite possible, however, that the other medicines you are on are constipating you and keeping you from having the diarrhea that you would otherwise be having.
But I'm disinclined to think that you are having gall bladder problems. While pain on the right side is usually a good indicator, it's also common with IBS and trapped gas. And while gall bladder pain can radiate, it usually stays on the right side or at least in the upper half of the torso; so moving to the back left side would be fairly odd. And as I said diarrhea and vomiting, not constipation, are the hallmarks of gall bladder disease, but, again, that could be because of the medicine you are on. But my main thought that this isn't your gall bladder is the fact that you feel better when you lay down and that you sleep well. I have not yet met a person (although I'm a firm believer in that anything is possible) who can lay down when their gall bladder is acting up. Laying down always makes the nausea worse, not better, and most people with bad gall bladders will sleep sitting up or in a recliner or walk around until the nausea passes. And waking up in the middle of the night terribly sick is common for gall bladder suffers (although not a requirement).
I think you need to go back to your doctor--the one that's treating your thyroid--and tell him that you are experiencing constipation and that it got worse after upping your dose. He may need to switch you to another medincine. The constipation would explain too why you have a lot of trapped gas. Have you tried OTC remedies, such as Gas-X or chewable gas relief pills or Beano?
Have your doctors done blood tests to check for cancers? Yes, ovarian cancer can look like IBS, but I think it tends to cause diarrhea, not constipation, and pain in the lower abdomen, but don't quote me on that. If your tests come back clear for cancer, then you should stop worrying; everything you describe is typical of IBS and/or ulcers.
Oh, and did your doctor talk to you about the new information that links stomach ulcers to a bacteria rather than an overproduction of stomach acids? (I'm betting not since he has you on an antacid.) There has been some evidence (none conclusive yet) that ulcers can be treated with probioitcs (like acidophillus and yogurt cultures). You may want to try a cup of yogurt a day to see if that helps with your ulcer. Calcium can be constipating, so make sure it doesn't make you worse, but on the up side calcium seems to help repair and de-irritate the lining of the stomach and intestines, so it's worth a shot. If the yogurt does bother you further, then you can always try just plain calcium pills or pills with multiple good bacteria cultures in them like Culturelle or Digestive Advantage.