i have french tarragon and it starts to go dormant this time of the year, i do prune off the brown stems to about 2 inches , also harvest the fresher looking ones and dry them until the new shoots start up. use it on fish or chop some up when i make chicken salad, ina garten from food network has a good recipe for chicken salad using tarragon,or infuse tarragon in vinegar for salad dressing . when the pot gets crowded i divide them into more pots, while dormant. we don't have snow or hard freeze here in san francisco , so the plant stays out.

It comes back from the roots every year and will spread, though not aggressively. About all it needs is weeding.