Could be two things:
Spider mites like Dracena Massangea that are located in dry conditions like the inside of your home or an office. (Dry is relative humidity < 40% for tropicals). They can be easily detected by examining the underside of the leaf in good light. You could see small white specks. A magnifying glass will allow a definitive diagnosis. Severe cases have webs usually located in the axils of the newest growth.
A solution of dish soap and water applied by running a cloth over the underside of all the leaves at about 6 day intervals for 3 or 4 repetitions should control or eliminate them.
The most common cause is overwatering and consequent root rot. This does not look like the case here.
EDIT Root rot starts from the soil and works upward. New stalks can continue to look green for a while until the last of the available water is used. The final symptom is often papery dry leaves at the point of newest growth. A good test for root rot is to grip the trunk and see if it the bark is firm or has softer hollow areas that run vertically. As root rot progress vertical sections of the trunk stop moving water up.
However the last cause is competition amongst the trunks. These trees are usually grown in Costa Rica then cut down with chain saws and shipped like logs to Florida for finishing. The smaller trunks can be out competed by the larger ones. Here, the only solution is to move the plant to a sunnier location and rotate the pot once a week to ensure even lighting.
Your last question is about rooting cuttings. This is ridiculously easy for a healthy plant in good light. Cut a section of new stalk off that has at least 3 or 4 inches of bare stem and place in water in good light.
A plant with rot or mites is not going to have cuttings that do much better so you need to diagnose the primary issue first before taking cuttings.
In my ten years as an interior landscaper I noticed the most long lived massangea were the ones planted in a clay soil. This is totally contrary to what you would expect but an interesting idea if you have clay soil and a healthy plant. The dangers of microbes and "bugs" in soil from outside are exaggerated in my opinion.