As you’ve now discovered, sometimes plants have one section that dies (deadwood) while the rest is green. This is common in all perennial plants. Remember, sometimes a section simply needs a little more time to bud. If section looks dead but still has give and green inside the stems, wait a few weeks to see if any green buds begin to form. If they don’t, go back to step two and repeat.
Once you’ve determined that a section of the plant is dead, it’s best to remove it. Simply cut the deadwood out with clean shears, doing your best to avoid cutting or damaging the live portions of the plant (lavender is hardy so don’t be fearful – just mindful.)