I don't have much to offer in the way of advice, except to be vary wary of a coyote or coon or about any other wild animal that would allow you to get close enough to engage in hand to hand combat. That isn't normal and is an indication of an animal that is sick. Normally people think of rabies, but canine distemper is more likely. NEVER EVER get close enough to an animal like this you can touch it or allow it to bite you.
First, so sorry for you loss and such a traumatic experience!
I have to agree with Howard E. I too live in WA state and had experiences around coyotes, both on the East and West side of the state. I have noticed the coyotes on the west side are not nearly as human afraid as they are on the east side. I think most coyotes on the west side are urban so they are way more used to people, plus there are the idiots that decide to feed them which makes the situation worse.
That said, a coyote should still run from a human that is approaching it, let alone get close enough to physically battle it. Something is just not right with a wild animal that will allow that unless it is possibly defending its young.
In our case, we live rural on the west side of the state and are well armed and know how to use our guns so we always have one at the ready for the type of situation you described.
Since you're not a good shot, there have been many other good suggestions made on this thread that should work.
Have you talked with the neighbor whose calves have been attacked? Maybe they are willing to deal with the coyote since they too have a stake in its demise?
I'm also wondering that if you were to trap it live and take it to Fish and Game, wouldn't they most likely just let it go somewhere else? Would they actually dispatch it as a nuisance animal?
Again, so sorry for you loss and don't ever feel bad or guilty for mourning a pet. We feel what we feel when we experience the loss of a beloved pet because they so often really are part of the family.