I definitely agree with OFL and 1f2f.
The reason your water is cloudy and your fish are ill is due to three main reasons:
1. Water quality. Fish constantly excrete ammonia through their gills as waste--this ammonia quickly builds up in small containers and is extremely unpleasant for your fish. Ammonia is corrosive--it's the unpleasant smell in most household cleaners, and it can easily burn your fish if you allow the level to get too high.
The only way to safely manage ammonia in small tanks is through frequent 100% water changes. If you change 50% of the water today and 50% tomorrow, technically this adds up to 100, but that doesn't mean you've changed 100% of the water. There's inevitably some of the old water left over, plus whatever your fish has added to it in between--so over time, doing only partial water changes becomes less and less effective as the concentration of ammonia being left behind increases. I personally do not buy into the idea that changing your water once a week is effective at all--I would be changing 100% of the water every other day in a tank that tiny.
2. Temperature. Bettas are tropical fish that need consistent temperatures around 78-83 degrees to feel comfortable, healthy, and active. Since they are cold-blooded, their entire metabolism is dependent upon temperature. At room temperature, many bettas are lethargic, sickly, experience a loss of circulation, and digestive problems. Unless you have a dedicated fish room that is set to 80 degrees, you will need a submersible heater to help sustain the necessary temperature. Most quality submersible heaters are designed for use in containers that are a minimum of gallons in size--this is a big reason why many believe that 2 gallons is the bare minimum acceptable tank size for bettas.
3. Environmental Enrichment. Bettas, just like other animals can feel bored, scared, and stressed. They have the need to express their natural behaviors and exercise. It is very difficult to provide your fish with relief from boredom, places to hide and feel secure, and space to exercise and patrol his territory when the tank is small and relatively bare. Many bettas in these situations develop destructive neurotic behaviors such as glass surfing and tail biting. Others become obese from lack of exercise--a very common cause of death among bettas.
Honestly, the best way to remove the burden of constant maintenance and improve the lives of your fish is to upgrade to bigger containers. This doesn't have to be expensive--you can easily get rubbermaid/sterilite containers from walmart for only a few dollars. These containers can be heated and provide your fish with lots of swimming space.
I think your ultimate goal should be a ten gallon divided tank for both of your fish to live in. This will be the most cost-effective way to care for your fish, and easiest to maintain. One large tank will take up less space than two smaller tanks, will be more stable, and you only have to maintain one tank. Easy. Sometimes you can find an entire ten gallon setup (heater/filter/light/decor) for $20-$30 or even free on craigslist/freecycle.