My experience in NJ has been that lavender wants sun and lots of it and seriously excellent drainage; they'll take crappy soil if the drainage is good. I would check the drainage under the oak tree - if the soil is heavy, you may need to move the plants.

Feeding will not make up for the lack of sun, may make it worse as Lavender also does not like to be fertilized and does not do well in moist/wet soil.

Lavender needs full sun and dry well drained soil. You will need to remove trees or find a different plant with the same colours that will grow in shade.

The lavender might do better in clay pots, but would still prefer full sun.

Salvia might be a substitute if you get at least 4-6 hours of direct sun.

Here is a link that might be useful: Salvia nemorosa 'East Friesland'

I am sorry, but lavender dies in the shade. Some people surround their plants with reflective surfaces to allow the leaves get more light. Well, it does not work with lavender. (It helped me with some annuals, just to get them going.)