Congratulations. I have looked through a Binuxit in very good condition and liked it very much indeed. Good brightness for a 1960s binocular with single coatings, I liked the colour rendition, and the overall feel and controls had that "old world" feel I enjoy. I hope your astigmatism does not detract too much from your enjoyment of what I think is still a fine glass. I'm not sure how much this may help - but it could be worth experimenting with opening and closing the IPD (the distance between the barrels) a little to see whether placing the binoculars in just the right spot over your eyes may help.
I'd say a binocular like that deserves to be looked at by someone competent - I have sent binoculars to East Coast Binocular Repairs in Norfolk whom I can recommend, but I'd be surprised if there isn't at least one good repair shop closer to you. Regreasing and relubricating the mechanicals and cleaning the lenses/prisms is routine for one of these gents, but not a procedure I'd like to take on myself; and although old Binuxit models are said to be pretty clean inside, the fact that yours has fungus indicates the surfaces of the lenses/prisms may have some haze. It wouldn't surprise me if they had not been serviced in their lifetime, and 50 years or more is a long time for any optical device to do without.
Lens caps should be obtainable from most binoculars shops - best thing is to take yours in so a perfect fit can be matched. Eyecups - short of having replacements made to spec by a machinist, or filing down the existing ones (not a job I would care to try myself), I can't see any other options.
Don't forget the Swarovski Habicht was (is) made over a much longer time frame than the comparable Leitz or Zeiss 8x30 porro prism models, and will have benefited from very much improved multi-coatings. If your father's Habicht is a new one you should notice a distinct difference in brightness, but your binocular I still feel is very functional.