Let me see if I can get everybody on track here.
Maybe myself if I am the one wrong.
A real common problem when you switch to low current bulbs, is that they will not flash, the indicator will just stay on constantly.
That is the problem some folks here are trying to help you fix.
If that is what is happening, then I am wrong, and you need to listen to them. Or my corrected ideas when you confirm I am wrong.
BUT, from what I am reading, the lights flash OK, and the indicators flash with the lights.
The problem is, the indicators stay on when you turn on the headlights without the flashers going.
If so, then that is usually a bad ground. You said that there are not two elements, but there does not have to be.
You have two groups of LEDs with a common ground. That is the same as two elements with a common ground.
If that ground is not hooked to ground, then the marker light voltage will feed back through the turn signals looking for a ground and will find it through the indicators.
You really need to eliminate a variable here. Before you do a lot of work, put the original type bulbs back in temporarily to see if the problem goes away.
If it does, then we know to start looking at the new bulbs and how they are hooked up.
If not, then we can start looking for the actual problem without the distraction of the new bulbs.
For instance, the rear lights will cause the same problem if you have a bad ground there.
Also, since the E-flasher uses a separate flasher module, it might have to be swapped out due to low current.
And, don't forget a bad or stuck brake light switch can cause the E-flasher to stay on without flashing.