2006 Chevy Cobalt power dying with new battery in it, is it the starter?

Generally speaking, if the starter was dead it would not jump start.
Sounds like the key switch or the neutral safety switch.
You would need to know if the starter has power when the key is in the start position.

Could be a weak connection between the battery and starter, or the clutch safety switch. You would need to know how far power is getting in the starter circuit to figure it out.

Park your Chevy Cobalt on a level surface, activate the emergency brake and place blocks behind the back tires for safety. Disconnect the negative battery cable with a terminal puller. Raise the front end of the car with a jack for easier access to the starter.
Remove the intercooler pump's outer bracket in models with a 2.0L engine. Place the bolt, and all other nuts and bolts, into the jar for safekeeping until later.
Disconnect the starter electrical connections, including the S-terminal and positive battery cable. Label the wires if you have concern about remembering where they go when you install the new starter.
Unscrew the starter mounting bolts and remove the starter from the vehicle's engine.
Take your malfunctioning starter to the parts store with you. Aftermarket distributors remanufacture most of the starters they sell, so you can likely obtain a partial credit toward your new starter by exchanging your old one. Be sure that you take care not to damage the starter in transit, because the better condition it is in, the more you can get for it.
Purchase a replacement starter that has the same or higher cranking capacity (power rating) as the original starter. Also, match the bolt patterns, drive gears and electrical connections. Your owner's manual may list specifications or the auto parts technician will help you determine them. Download a free owner's manual by registering with the Chevy Owner Center (see Resources).
Install the new starter by reversing the removal process. Torque the starter mounting bolts to 37 foot pounds (50 Nm). Tighten the bolt for the intercooler pump bracket to 15 foot pounds (20 Nm).
Lower your Cobalt back to the ground. Reconnect the negative battery cable. Turn the key in the ignition to see if your new starter motor functions correctly.

Parasitic draw is what's happening . Have see instrument clusters cause this ,as well as module's not powering down . You need to do testing to find the problem . Watch these videos an test .
Diagnose Battery Drain Parasitic Draw Chevy Trailblazer
The Trainer 27 3 Ways Of Finding The Cause Of Key Off Battery Drain...

I'm sorry to be the one to have to tell you this, but you are going to need a new engine. The connecting rods have welded themselves to the crankshaft, due to oil starvation. That is why it will not turn over, even with a new starter. I would take that starter off and return it for a refund. Just make sure it's clean and doesn't look like it has been installed'

There can be a couple things, nutral saftey switch, AC DELCO, NEUTRAL SAFETY SWITCH Direct Replacement Part For OE Number 24219476 `$68.67 , ignition switch not adjusted correctly, try jumping the starter at the starter to see if the starter is working, of course I have just fixed a car that the guy didn't have the shifter in park, try these things and get back to me

Try new spark plugs and wires, or an overall tuneup. This is a fairly new vehicle, shouldn't be to much wrong with it, unless it just haven't been taken care of.

2006 cobalt won't start after a severe thunderstorm. Battery was check and is good, light turn on, turns signals work a/c blower works, however, radio does not.Doesn't even click or turn over. All seniors check out. What's the problem?

Sounds like your battery is dead. Most batteries last about two to three years. I suspect maybe yours has a couple of dead cells. You will need a hydrometer a device we use to test each cell. When you pick one up at the parts store pry open the caps off of the top of the battery. Next squeeze and hold the bulb on the hydrometer and stick the meter into each opening in the battery and release and make a notation of where the little ball in the meter is located. This will tell you that you have one or more cells dead and the battery will need replacing. Good Luck.

so my 06 cobalt keeps dieing and i have to jump it alot lately is it my alt or my battery