Yup alternator. Had that happen on my 03. The reason you made it so far may be that the alternator itself is still charging. These alternators have a a separate circuit for the gauges and warning light. This can malfunction even if the alternator is still working. That's what happened to mine. It still requires replacing the alternator though.
Bryan
2003 2500HD Ext. cab short box
6.0 liter 4.10 gears, Nelson performance PCM 293,000 miles
98 K1500 4x4 heavy duty 1/2 ton (Sold)
6,600lb GVWR 5,280lbs on the scale empty
14 bolt rear diff. 3:73 , Tranny and oil coolers
380,000 miles.