Answer: Lavender is almost a woody shrub, especially as it gets older. From your description it sounds as though you might be able to layer the branches that are so close to the ground by heaping a bit of compost and soil over top of the part that is touching the ground and thus encouraging it to root. Otherwise, tip cuttings root fairly easily in dampened soilless mix or sand.
In terms of pruning, cutting into the oldest wood can shock the plant and kill it. A light pruning of the new growth now in late spring will encourage the subsequent growth to be dense and bushy and thus cover up the old bare stems.
Regular pruning can help the plant look better over time--a trim or shearing in late spring along with removal of any winterkill, followed by a shearing after each bloom flush should help keep plenty of vigorous new growth coming in. It sounds as though your plant may simply be showing its age and could be replaced.
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