Hmm... well all that can be said seems to have been said... Speaking for myself, if I were in your shoes, I would:
Keep rifle larger than .22 handy, unless you are not trained and comfortable or accurate with it.
Use 00 shot in shotgun... if you must use a shotgun... but range is very limited.
Visit neighbors and let them know you are concerned about coyotes or hybrids or wild dogs in the area.... don't threaten, just let them know you feel very threatened, personally. Observe their dogs/pets so you can avoid targeting them if possible... but hopefully they are being restrained to their home territory by your neighbors... if not, then you are now aware of that. I would agree with the comments of others that the behavior you describe seems uncharacteristic of the coyotes that I am familar with... however, very common for packs of free ranging domestic dogs doing baad things.
Obtain snares for coyotes and large enough for dogs, since you could get either. I purchased 12 heavy snares at the local ranch supply store for $29. Learn to set them where you see the obvious scat and trails... and, very important, run your snare line very early in the morning.... and, observe the numerous S's... snare, shoot, shovel, silence.... you will catch something... maybe what you are after, maybe not... if you set the snares only on your property and only where you SHOULD catch your tartet animal, that is the best you can do. However, if others know you are doing this,they will attribute all strange occurances and disappearances to you... and some will not be happy about it... so... shhh...mum's the word...and, you'll have to determine the local laws re taking coyotes. In Texas, they may be taken all year round. Also, dogs on your property may be shot if you personally or your property (livestock, pets, etc.) are in immediate danger because of them.
You can seek to hunt them.. I'd start in daylight... sitting in a stand... and be aware that they are extremely wary and have superlative noses. I wish you success but predict you will spend a lot of time observing nature without seeing your target. A rabbit call at night can be effective but again position yourself carefully and use a red headlight for spotting. Again, success will likely elude you.
Here in Texas snares seem to be the most effective at actually catching coyotes. They can be set either with or without bait. There is information on the net about how to set them....PM me if you can't find it. They work day and night... forever until they catch something...if you leave them set that long...They are the most effective, low dollar cost, least time comitted approach that I know of.
Be aware that you will not eradicate whatever is bothering you. You may get some, maybe even all of the pack you have heard but not seen yet... but more will move in... either coyotes or dogs or hybrids... so,this will be an ongoing activity. Be prepared for that... and adopt the attitude that this gives you a reason to wander over your property at odd times doing odd things.... and, remember, you and your neighbors will be happier if you simply do your deeds without comment, bragging, fanfare, or mentioning it at all.... you may even decide to keep your wife in the dark regarding the extent of your activities..... if you are successful in your efforts, the instances of concern will reduce and hopefully your wife will decide to stay in the neighborhood with you.
Good luck!!! But, don't tell me about it!!!