Intermittently, when my engine is running, all turned-on lights will pulsate (brighten then dim) repeatedly?

Yes, this can happen when the ground from the battery to the chassis or the ground from the battery to the engine is no good. I would run a jumper wire while the engine is running, and the lights are on to see if it makes a difference. If not, the alternator is bad and the auto parts store can't tell. Here is how to replace the alternator correctly in case you need it.

ALTERNATOR

Removal and Installation

REMOVAL

1. Disconnect the battery cable from the negative terminal.

CAUTION:
To prevent damage to the parts, disconnect the battery cable from the negative terminal first.
2. Remove air duct (inlet). See: Air Cleaner Fresh Air Duct/Hose > Removal and Replacement > Exploded View.
3. Remove reservoir tank. See: Radiator > Removal and Replacement > Exploded View.
4. Disconnect alternator harness connector.
5. Remove "B" terminal harness nut, and then disconnect "B" terminal harness.
6. Remove alternator mounting bolt (upper).
7. Remove engine under cover. See: Interior Moulding / Trim > Removal and Replacement > Exploded View.
8. Remove front wheel RH.
9. Remove splash guard RH. See: Front Fender Liner > Removal and Replacement > Removal And Installation.
10. Remove drive belt. See: Drive Belt Tensioner > Removal and Replacement > Removal And Installation.
11. Remove idler pulley. See: Drive Belt Tensioner > Removal and Replacement > Removal And Installation.
12. Remove compressor mounting bolts, and then move compressor to secure work space. See: Compressor HVAC > Removal and Replacement > Exploded View.

CAUTION:
Never disconnect low-pressure flexible hose and high-pressure flexible hose from compressor.
13. Remove water pipe mounting bolts, and then move water pipe to secure work space. See: Oil Cooler, Engine > Removal and Replacement > Exploded View.

CAUTION:
Never disconnect water hose from water pipe.
14. Remove return tube fixing clamps, and then move return tube to secure work space. See: Power Steering Line/Hose > Removal and Replacement > Exploded View.

CAUTION:
Never disconnect return tube from return hose assembly.
15. Remove alternator mounting bolt (lower) and alternator mounting nut (lower).

16. Remove alternator from the right side of the vehicle.

CAUTION:
Be careful not to contact with and damage surrounding parts when removing alternator from the vehicle.

INSTALLATION

Note the following items, and then install in the reverse order of removal.

CAUTION:

- Temporarily tighten all of alternator bolts and nut. And then tighten them in numerical order shown in the figure.
- Install alternator, and check tension of belt. See: Drive Belt > Procedures > Checking Drive Belt.
- Be careful to tighten "B" terminal nut carefully.
- For this model, the power generation voltage variable control system that controls the power generation voltage of the alternator has been adopted. Therefore, the power generation voltage variable control system operation inspection should be performed after replacing the alternator, and then make sure that the system operates normally. See: Charging System > Initial Inspection and Diagnostic Overview > Inspection Procedure.

Check out the images (below). Please upload pictures or videos in your response to the problem so we can see what's going on.

It will only be the ground wires to the engine/transmission and to the chassis to the battery, I would check those first. These need to be clean and tight for a good connection.