battery light is to let you know that the battery is not being charged enough. Yes, could be the alternator. an alternator lists for around $160 for a reman to $465 for a new one. That's list, you might be able to find it cheaper if you buy it yourself. Labor is .6 hours so, whatever the labor rate is in your area. based on the 5.3L engine. and a 105 amp alternator. you could have a 145 amp or even the 200 amp alternator. have to see what alternator you have.
check engine light is on to tell you that the computer has a problem with something that will affect your emissions and/or driveability.