yah its a good possibility with ur old alternator dying, im sure it discharged the battery, and now it doesnt have enough amps to run for a long amount of time, without it being fully charged itll keep running down, and although the alternator does charge the battery it is meant to only charge it while the car is running, and only a little at a time, not meant for fully charging the battery, id strongly suggest making sure the battery is in good condition and fully charged, ive had batteries be bad that were only a month old so get it tested first if u ever have electrical issues.