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It sounds like a fungus issue, which can be alleviated with a fungicide. Neem oil is a good one to try.

After a time, lavender tends to get very woody at its base. It is not as likely to have new growth on very woody stems. The stems closer to the top are not as woody, so they will produce new growth better.

You may want to consider either replacing the whole plant or starting a regimen of rejuvenation pruning where you remove the oldest stem or two each year to encourage new younger stems to grow from the base.

It is a true Lavandula (lavender). So, you should be able to use it in all ways a typical lavender can be used, including culinary.

You can separate the plant at the ground level. Even if it is woody, as long as the stems at ground level are two separate stems and they have their own root system when split, they can be divided.

This normally happens because the plant do not get quite enough sunlight and they have weak stems. If you can get them more light, that would be helpful. Too little water is another factor.My daisies do this every year when the weather gets hot. After a good soaking, they usually perk right up. It may also help to prune back the plants so that they can produce more bushy growth.