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Transplanting infected plants will transplant the fungus or virus as well. There are several diseases of black-eyed susans that cause blackening leaves, both fungal and bacterial. One is devastating to the plant, others are more of a cosmetic issue. It is safer to divide a healthy plant this fall (or move one of the numerous seedlings if you let a healthy plant go to seed) than transplanting a diseased plant. Here is a thorough description of Rudbeckia diseases: https://www.extension.umn.edu/garden/yard-garden/flowers/diseases-of-rudbeckia/

Your plant appears to be suffering from "aster yellows". It is an infection typically spread by leaf hoppers. If you are within a month of your first frost, I'd cut the plant to the ground and make sure every bit of it is tossed out. It might be a mite, too. Here is comprehensive info: http://www.missouribotanicalgarden.org/gardens-gardening/your-garden/help-for-the-home-gardener/advice-tips-resources/pests-and-problems/diseases/viruses/aster-yellows.aspx

It would definitely be a good idea to remove the dead blooms, and any other dead growth. Here is an article for more information: https://www.gardeningknowhow.com/ornamental/flowers/black-eyed-susan/deadheading-black-eyed-susans.htm

Seeds slowly develop at the base of the flower, usually in a pod. It takes several weeks for seeds to mature. The pod turns brown when they are ready. If you don't have any seed pods, it may be you have a sterile cultivar or the flowers are not being pollinated. Have you looked at google images of Thunbergia seed pods?

It looks like you are in gardening zone 5b. Black eyed susan vine only survives the winter in much warmer climates (zones 10 and 11). It is possible to keep the plant over the winter by taking cuttings of it now, and growing them indoors until next spring:

https://www.gardeningknowhow.com/ornamental/vines/black-eyed-susan-vine/black-eyed-susan-vine-care.htm
https://www.gardeningknowhow.com/garden-how-to/projects/rooting-plant-cuttings.htm

do you have a vine black eyed Susan the vine ones are annuals.
perennial Black-Eyed Susan is hardy, especially if you give it a light mulch of dried leaves—mimicking the way leaves would catch in the flower stalks if the plant grew wild. Annual Black-Eyed Susan self-seeds, so new plants will appear in the spring. Keep the surrounding ground bare, so seeds have a place to rest through the winter and sprout in the spring.

If you cut back stalks of perennial Black-Eyed Susan, wait until late fall when the plant is completely dormant. Leave three or four inches of the stem above the basal leaves to avoid injuring the plant. Cover the plant well with a mulch of dried leaves, especially in the first year after planting or dividing.

If you cut back stalks of annual Black-eyed Susan, it will not be able to self-seed. Lay the stems with the seed heads on a paper plate and let them dry. Shaking or rubbing the dry seed head will release the seeds onto the paper plate. You can store them through the winter in a labeled envelope or plastic bag and plant in the spring.