I've got good news for you! This is totally normal and your plant looks healthy and your watering practice sounds very right. 3 weeks? You must have trained your plant to be drought tolerant by watering only when necessary. The roots have gone deep and are able to reach the last drops of moisture available!
Leaves die off periodically until the entire plant dies. Normal and healthy. Just cut them off as soon as you see a leaf that is starting to be cut off by the plant itself...mainly because of old age and/or bad positioning to get light. The sooner you help your plant prune itself more energy is available for the entire plant. All plants that have leaves not able to produce enough food and costing the plant more in energy to maintain than the energy made, will expend a bit of energy to cause abscission, cutting the leaf or branch off from water and chemicals and allowing the non or low productive leaves or branches to drop.
When we humans remove top soil and organics or even denude forests of long term existing vegetation and when we want to put plants in pots we humans have to add certain chemicals plants need so they are able to make their food through photosynthesis. Less is far better than more. Just a wee bit too much and one can quickly kill plants. Fertilizer IS NOT FOOD.
I prefer Osmocote for newbie gardeners and indoor plants. Extended release, 14-14-14 will pretty much last half a year. Great for foliage as well as flowers/reproductive stuff. I'd buy bottled water for indoor plants as our tap water is full of 'salts', fluoride, chlorine...on and on (shouldn't be drinking tap water yourself, unless you are on a well). You'll start seeing browning of the leaf tips and margins. Always use potting soil for potted plants. Seriously a big deal. Your instincts sound very good!
One other detail, if you want your plant to get more branching you need to cut the 'apical' or 'terminal' bud off. The very very end of a branch or at the top of the main branch at the top of the plant. Most of the energy in a branch of a plant is in that bud. When one cuts that off (called heading) that energy is then redirected to all the lateral buds down that branch. This is what one does for shrubs and hedges. This is not what one normally wants to do with trees unless one wants a bush on a stick.