DIYer advice: Your post isn’t very clear. But here are some points:
- A bad battery will not cause the engine to overheat.
- If you have the 4 cylinder 2.5 engine, a slipping serpentine belt will cause the battery to go flat (because it’s not turning the alternator), and cause the engine to overheat (because it’s not turning the water pump). (I don’t think this applies to the V6 3.5 engine.)
- " started smoking and overheating". What started smoking? How do you know it was overheating (did the temperature gauge on the dash go up into the red zone?)?
- Have you checked the coolant level? Is it low?
- Have you checked the engine oil level, on the dipstick? Is it low?
If the alternator goes bad, and isn’t charging the battery, the red “battery” warning light will usually come on while the engine is running. Plus, after a jump start, the battery will go flat again, and the car will stop running a few minutes after you disconnect the jumper cables. If you didn’t see that warning light, and if the car continues to run OK once it’s been jumped, the alternator is probably OK.
If a car battery goes very flat, that tends to damage the battery, so it can’t hold much of a charge in future. Plus, batteries often don’t last many years.
So it’s probably the battery. But places that sell batteries will normally test your battery and alternator for free, and it’s best to get them tested, just to be sure.