Putting enough lime into the soil to kill disease organisms is also likely to temporarily sterilize the soil, too--eliminating any gardening there for the next year or two. Powdery mildew , which is what is affecting your peas, is an airborne disease, anyway.
I would increase the diversity of your soil's ecology by applying compost, 48 hour compost tea, vermicompost, and/or fish tank water. That will have two good effects: disease organisms will have serious competition from beneficial organisms, and it will stimulate the plants' immune systems, helping them to fight off pests and diseases more easily.
A little bit of lime may be beneficial in balancing the acid produced by all that organic matter. If that is needed, test the soil pH first to be sure of how much lime is needed, and use a milder form, such as agricultural limestone, since that is friendlier to soil organisms.

I agree with Tug. I would hold off on the lime and use a well balanced organic compost. My plants love the Alaska Fish Emulsion Fertilizer. It's just emulsified fish but has everything a plant needs including trace minerals. Kelp is also a good choice or simply cover the bed with a blanket of fallen leaves and till them under in the spring. Too much lime can affect the soil Ph which will inhibit plant growth. It's naturally very abundant in the soil already and seldom need supplementation - the calcium & magnesium.

I don't think you have a problem with your soil at all. It's quite normal for plants to look ratty at the end of the harvest season. That's because they've completed their life cycle and have expended all of their energy on seed production. Crop rotation is a good idea and the microbes will fix the soil again.

A little lime at the individual plant stations fairly close to the plants when planting brassicas though is often beneficial in that it tends to ward off club root. A small teaspoonful is plenty!(Don't add it at the same time as any manure though)

As Bath says, peas and beans are going to die off after cropping anyway and do look pretty tatty. Don't pull them up, break them off at soil level so the nitrogen nodules on the roots remain to benefit next year's crop (probably brassicas)

You can get pH metres now that you just poke into the soil and read off, without all the messy test tubes etc. Well worth investing in.