Sounds like the engine's battery is too low in charge to operate the vehicle properly, or does not have enough power to sustain a load.
Have the battery checked for state of charge and capacity.
If the battery is low, but not bad - then there could be another problem, such as a charging system failure.
If the battery tests good, then there could be some other electrical or starting system problem -- but I'm definitely thinking battery first.
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www.repairpal.com/repair-shops
RepairPal Certified shops can diagnose and repair your 2007 Chevrolet Cobalt. They provide transparent pricing with a minimum 12-month or 12,000-mile warranty on repairs and services.