1999 Chevy Blazer - Alternator?

My wife's Chevy Blazer has been acting up lately. About a year ago, we started hearing a clicking sound under the dash, kind of like the blinker was still on. It wasn't constant. Some trips it clicked the whole time, other times we didn't hear the click for weeks.

We didn't think much of it as it ran fine. About six months ago, it didn't start. I checked the battery connections (new battery, by the way) and found some corrosion. I cleaned it off and tightened up the connections. It started fine.

Two months later, it did it again. No corrosion this time, but after tightening, it started up.

Over the last month, it failed to start numerous times. Finally, last night, it didn't start and wouldn't start without a jump. When I jumped it, the Blazer ran just fine until I disconnected the jumper cables. After disconnecting, it died out right away.

I suspect the alternator. Any thoughts?

Also, could the clicking sound be part of the problem?