Why do the headlights alternate on and off, left and right?

I'd check the headlamp relay in the under hood fuse / relay box to start . The relay is energized by the BCM - body control module !
Voltage is available to the headlamp and panel dimmer switch at all times. When the headlamp and panel dimmer switch is activated, the two following actions occur via CKT 352 (WHT):
?€¢
A headlamp enable request is sent to the body control module (BCM)

?€¢
The headlamp relay is activated

The VE1 option is equipped with automatic headlamps, which is activated when the photo sensor detects dark conditions. A code will set if there is a fault in the photo sensor, or in the two CKT's between the photo sensor and the body control module (BCM).
The coil output of the headlamp relay travels through CKT 250 (BLK) and is grounded at G101.
When the headlamp relay is activated, power is supplied to both the LH and RH headlamp fuses through CKT 1346 (LT BLU).
LO and HI beam functions are activated/determined by the multi-function switch on the steering column.
The HI beam function must be enabled by the body control module (BCM).
The headlamp power travels through the multi-function switch at the steering column and is grounded through CKT 750 (BLK).
The flash-to-pass function is also activated by the multi-function switch on the steering column. It is powered by CKT 1840 (ORN) and enabled through the body control module (BCM) via CKT 307 (YEL).

as you checked already fuses, relays and bulbs there must be something wrong with the light switch in the dashboard.
- either a loose connector or worn out contacts. first remove the switch and spray some WD-40 or similar contact spray into the connector from behind - if that does not help you have to replace the light switch with a new one...

The fuse block the relay is located in is behind the knee bolster, the trim panel under the steering column, left of the steering column underneath the dash. Common source of problems if that does not fix it is the Light Control Module ( LCM), turn signal switch unit, headlight switch.

I looked in the Honda technical manual wiring diagrams and I don't think there is a separate relay for the low beam headlights. The headlight switch itself is located right above the steering column where the headlight stalk is attached - controls parking lights, high beams and low beams. It could be that a relay is integrated within this switch,

check headlight bulbs all headlight bulbs

Hello, try to start by checking the headlight relay and headlight fuse. You also may want to check to make sure the bulbs aren't blown out. If those parts check out to be fine and in working order, you may need to replace the turn signal assembly in the steering column. This part controls the hi/low beams. This may sound crazy but, when GM wired their older trucks, they wired the headlights to go from the battery, to the hi/low beam switch, then to the headlight switch (to the left of the steering wheel) then out to the light bulbs. You can kind of test to see if this part is good by making a fist and (with your headlights on) gently hitting the plastic cover for your steering column that is right behind your actual steering wheel. This could cause your headlights to flicker on and off.

Thanks

There's only one relay, but two fuses, both located in the underhood fuse/relay box.

Sounds like you have a wiring problem since sometime the left works and the right doesn't, then the right works then the left doesn't. Might be the multi-function switch on the steering column in combination with the connector M10-1, EM01, SE14 & 16.

 I’m inclined to believe you have a headlight switch problem and here’s why. There are 2 fuses one for each of the headlights on the vehicle. For the daytime running lights (DRL) to operate the fuse and wiring to the right head light has to be good, because it supplies the power for both headlights in the DRL mode. The fuse and wiring for the left headlight has to be good because the DRL relay needs power from that fuse to operate. The DRL relay in turn provides a path through the left headlight and wiring to ground. When you pull back on the dimmer switch, you are activating the flash to pass circuit of the headlight switch assembly. Because the flash to pass mode works you then know that the high beam wiring and headlight bulbs are in good condition. This also tells you that the ground circuit for the headlight switch is good. In many newer vehicles, the headlight switch doesn't provide power to the lights. The headlight switch has now become the path to ground to complete the circuit. The headlight switch and the high/low beam dimmer control are one assembly so when you replace one you get the other free. Such a deal! My coffee bet is on the headlight switch assembly.

I would start by checking the headlight relay and headlight fuse.  You also may wanna check to make sure the bulbs aren't blown out.  If those parts check out to be fine and in working order, you may need to replace the turn signal assembly in the steering column.  This part controls the hi/low beams.  This may sound crazy but, when GM wired their older trucks, they wired the headlights to go from the battery, to the hi/low beam switch, then to the headlight switch (to the left of the steering wheel) then out to the light bulbs.  You can kind of test to see if this part is good by making a fist and (with your headlights on) gently hitting the plastic cover for your steering column that is right behind your actual steering wheel.  This could cause your headlights to flicker on and off.  It's not 100% guarenteed that this method will work, but it may.  
Any questions, just ask!

Sounds like the head light relay is bad . Have you ever checked the headlight bulbs?