Green snot is definitely an indication of infection of the sinuses. You do not always have a headache or pressure with a sinus infection.
Irrigating sinuses with salt/soda solution will help clean the sinuses. My doctor approved the following:
1 cup boiling water
1/2 teaspoon plain salt
1/2 teaspoon baking soda
Dissolve salt and soda in the boiling water, stir until dissolved.
Strain thru a clean paper coffee filter.
Put in clean nasal spray bottles.
Allow to "COOL" before using.
If solution is too drying, add 1 or 2 drops glycerin to the mixture.
To use, lay on your back with head hanging off the bed.
Squirt a few squirts into each nostril, turn head slowly side to side, massage cheeks, say KKKKKK while getting up to prevent solution from running down your throat. Blow gently as needed. Do this twice a day.
You may still need an antibiotic to clear the infection.
Remember, yellow and green snot indicates infection.