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Any plant food that you would use on a vegetable garden should be fine, but Rosemary doesn't really need careful fertilization in order to do well. I routinely grow very healthy rosemary plants with no fertilizer at all. Soil composition and sunlight are really more important factors. These articles may help you:

https://www.gardeningknowhow.com/edible/herbs/rosemary/pruning-rosemary.htm
https://www.gardeningknowhow.com/edible/herbs/rosemary/growing-rosemary-plants-rosemary-plant-care.htm

Your plant has older or woody stems and the new tender growth is softer and pliable. This is a normal growing pattern of a Rosemary.

https://www.gardeningknowhow.com/edible/herbs/rosemary/rosemary-grown-in-containers.htm

The most common issue for a Rosemary dying is a watering issue; too much or too little.
Botrytis blight fungal infection rots the older leaves in the center of a rosemary plant. It flourishes in high relative humidity, poor air circulation, and cloudy weather conditions. Watch for yellowish brown, irregular leaf spots or water-soaked spots on plant stems. During humid weather, the fungus produces a fuzzy brown to gray growth on the surfaces of injured plant parts or dead plant material. When disturbed, the fungus emits a cloud of spores.

Remove and destroy infected plants and debris to control spread of the disease. Prune overgrown plants, and increase spacing between container plants to improve air circulation. Avoid mulching the plant beds with organic materials, because these harbor fungal spores. Instead, use crushed stone, pea gravel, or rocks, which dry out quickly and reflect warm sun onto the plants and reduce humidity still further.
https://www.gardeningknowhow.com/edible/herbs/rosemary/brown-rosemary-plants.htm
https://www.gardeningknowhow.com/edible/herbs/rosemary/white-powder-on-rosemary.htm
https://www.gardeningknowhow.com/edible/herbs/rosemary/pruning-rosemary.htm

Transplanting in the fall is best. Cut the plant back and water it before transplanting, try to dig up as many of the roots as possible, and follow these tips to increase your plant's chances of a healthy transition:

https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm

Sometimes transplanted rosemaries die, so you can also save the plant by taking some cuttings:

https://www.gardeningknowhow.com/edible/herbs/rosemary/how-to-propagate-a-rosemary-plant.htm

It sounds like powdery mildew or another fungal disease. This article should help
https://www.gardeningknowhow.com/edible/herbs/rosemary/white-powder-on-rosemary.htm

This would be anything that would provide warmth to unhardy plants to keep them alive. Unfortunately, Rosemary is only hardy in zone 9 and up... Which is the very southern portion of the U.S. You are well outside of the growing zone for these plants, and there would be no hope of getting these to survive in your area. The only thing I could suggest is growing them inside under horticultural lighting.

This article will give you more information about growing rosemary: https://www.gardeningknowhow.com/edible/herbs/rosemary/growing-rosemary-plants-rosemary-plant-care.htm