IMO, if you are going to keep fish, especially bettas in an unfiltered bowl/vase, you NEED to have a test kit. it may seem like a hefty chunk of money ($20-30 or $35 depending on location), but that test kit will last you at least a couple years before it expires, and it is imperative to have one.
water changes need to be done DAILY on a small unfiltered environment. especially when you are medicating or treating for infection. clean water and a pristine environment are the number one things to aid in healing a fish. when dosing medications, it is best to redose them after a large daily water change, because medications break down quickly in water and are essentially useless after a short time. whenever i have to treat for a bacterial infection, i do daily 80-100% water changes and re-dose daily throughout the whole treatment period.
back when i had bettas in bowls, i did an ammonia test within 2 or 3 days of doing a 100% water change. the ammonia was at 4 ppm!!! that ppm is lethal for any fish, including bettas. since then, any betta in my home gets a filtered tank of at least 2.5 gallons, with live plants.
the darting could also be a symptom of parasites. fish will 'flash' against surfaces when they are affected by things like velvet or gill flukes. shine a flashlight on your betta. if he looks like he is covered in a shimmery 'powder', then i would suspect velvet. if you see anything funny/worm-like at the edge of his gill plate, that could be gill flukes.