Have both the "battery and alternator" tested at a parts store, usually done free of charge there. It's likely to be one of them as the problem. Just make sure the battery connections are tight and free of corrosion as a precautionary check. BTW, I'm leaning towards a bad alternator because the warning chime goes off when idling and goes away when accelerating. The increased RPM's when accelerating is spinning the alternator pulley at a higher rate of speed, thus allowing the alternator to put out enough voltage to run the on board electronics and keep the warning alarm for the charging system from sounding. When you slow down or idle, the engine RPM's drop, the alternator pulley spins slower and the alarm goes off because the alternator cannot keep up with the voltage demands. Sounds to me is that your alternator is partially working and on the verge of crapping out totally/failing, and when that happens, your battery won't be recharged by the alternator and your vehicle will run on battery power only for only as long as the battery power holds out, which won't be long.