I'm going to take a slightly different approach to your question.
I think all the other posters have correctly chided you for feeling your father owes you something and your entitlement mentality; so I won't play "pile on the rabbit."
I also think that in commenting on that, many posters have missed that you are trying to be a responsible horse owner and make sure your horse is provided for, for which I commend you.
So correct me if I'm wrong, the situation is you thought your father would help with either insurance or vet bills, and he's backed out? And you have some money saved for emergency vet bills, but not the amount you'd be comfortable with?
I would say that if your horse's value is under $10,000., you need either the emergency fund or the insurance, not necessarily both. Here's why - basic mortality insurance on the horse requires that you pay for surgery, most commonly colic surgery, before they'll pay on a mortality claim. So if you have a $5K horse, the insurance company my require you to pay for an $8K surgery before they'll pay the $5K for the horse. The options are pay for major medical insurance on top of the mortality, which is quite expensive, or to not insure but to build up an emergency fund.
If the very worst happens, and your horse colics and you have to make a decision whether or not to send the horse for surgery, you will have to deal with that then. If you do not have the money to pay for colic surgery, or decide not to for other reasons, that does not make you a bad or irresponsible owner. Not all colics are good surgery candidates, and it doesn't make sense to treat some colic cases that way. Same thing with major lameness that impact long term soundness; you'll make that decision based on the circumstances, you don't automatically send every case to surgery.
You have enough money saved to pay the vet bills for a non-surgical colic, or for a minor or moderate lameness or a wound. That's a good start, and better than many horse owners.
What makes a bad or irresponsible owner is not being able to provide BASIC veterinary care, or allowing the horse to suffer because they don't have money for the vet. Unless I've missed something, that is not your situation.
I would focus your energy on continuing to build your emergency fund, and continuing to build your horsemanship. Insurance is terrific to have, but there are other options - it doesn't make sense for some horses and some situations. If you have a nice pleasure/Pony Club/local show horse as opposed to a serious show horse with a competition it may NOT be the best route.
The best way to deal with your father backing out of his promise is to be smart, independent and self sufficient. I don't think your situation warrants you selling your horse; just continue building your emergency fund as best you can.