Tarragon flopped and gone woody

My French tarragon did really well this year. I’ve ignored it, not cut enough of it, and the branches are now woody for at least half of their length, and have flopped. The tips still look OK, nice and green. My current plan is to cut back a couple of the branches hard. Then leave it until spring and repeat. Is that a crazy idea? Can you cut into woody stems of tarragon? I know some things might not come back if you do that.
Carmarthenshire (mild, wet, windy). Loam over shale, very slightly sloping, so free draining. Mildly acidic or neutral.
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I cut mine right back a few weeks ago and it's re-growing, but not with much enthusiasm - I can only guess it's weather related.
A pity as it's one of those herbs I really look forward to in the summer.
I moved it into the greenhouse a few days ago to see if that would help.
PS - my tarragon dies back completely over winter leaving no woody stems
Billericay - Essex
Knowledge is knowing that a tomato is a fruit.
Wisdom is not putting it in a fruit salad.