BettaObsession, Lupin is spot on with his all advice. He's one of the gurus here on AC.
The fish stores very often will say the water is perfect when in actuality there may be ammonia and nitrite levels.
Perfect water should have 0 ammonia, and 0 nitrites. Water quality is one of the major causes of dropsy, due to the damage that poor water quality can cause to the fish's organs.
You absolutely must get a liquid test kit to test for yourself. Since your Betta is sick it is especially vital. Test strips are unreliable and give false readings most of the time.
The API Master kit has vials with bottles that, added to you tank water in the vials, gives you precise readings of ammonia, nitrites, nitrates, and ph.
The lfs probably used strips to do the testing of your water and even if they accidently did get an accurate reading they often don't think ammonia at .25 or .50, or nitrites at .25 or .50, or even traces of those toxic byproducts is a problem.
Those readings would certainly be a problem and will make your fish sick in the first place, and even sicker now that he is showing obvious signs of illness.
I used to use the strips. They are cheaper up front, but per test they are WAY more expensive than the liquid kit. I have a kit that has lasted me months, and I test everyday, sometimes several times a day. I would run out of strips in less than a month.
I cannot state it strongly enough that you MUST get your own test kit. You have to act quickly by doing big enough and frequent enough water changes, with a good dechlorinator like PRIME, to remove chlorine, and chloramines, and Prime detoxifies ammonia and nitrites for a period of time (appx 24 hours) giving you some breathing room to do the needed water changes within that time frame.
Hopefully, your betta is just constipated, but even if that's all it is, he had got to have pristine water to help him get through it.
Bettas may be kept in cups all the time at pet stores, which is not good, and your tank is bigger, which is good, but they still need filtration. You need to get a little Whisper filter, or some other brand that is good for a small tank, or better yet, get a 5 or 10 gallon to put him in. Bettas just thrive and become so happy and full of personality when they have room to swim around.
They also like temps that are regulated, and a 3 gallon unheated can fluctuate fairly quickly with the room temperature. I'm not an expert on Bettas, but I believe I've read that they like it around 78 degrees in their tank. You can check on that.
First things first, though:
I would do daily big water changes for him, since you don't really know what your parameters are, and since you have no filter. At least 75-80% ( I would say 100% daily but I hate to stress him out too much since he seems pretty sick)
Get a good conditioner that also detoxifies ammonia and nitrites. I think Prime is excellent and very economical. WalMart doesn't carry it, but the pet chains and the local fish stores do.
Fast the fish for a few days.
Try the nuked, peeled, frozen english pea for constipation. If the Betta seems to not want to eat, get garlic (fresh or minced) mash it and get the juice and smear and mush up the pea with the garlic. Fish just love it and it is appetizing to them. (works for me... mmmm Italian food, lol)
We are all pulling for you and your little guy. You are his lifeline. He's depending on you to do everything you can, and sometimes it's not enough and we lose our little friend even with our best efforts.
Keep ups posted. We're wishing you success.