Synthroid is Thyroxine

Yes, they are the same T4 replacement.

You need to replace the missing T4 - supplements help, but they're not a substitute.

Bear with me for a sec...

Compare being hypothyroid as a car with flat tires - the car is still driveable, but inefficient (like a hypothyroid person). Add air (ie T4) to the tires and it becomes more efficient, but the right amount needs to be provided - too much or too little won't do.

Iodine supplementation is fine, but it will not replace the missing T4. If you can't tolerate the drops well, foods like eggs, cheddar cheese, fish, and sushi with seaweed all are Iodine rich.