A couple of things here -- or you will have a tragedy before long
. If she is this unreliable, she hasn't been adequately trained on recall (and doesn't come easy -- this takes daily effort and daily positive practice/reinforcement all through her life but should be fun to do for both
) -- have you done a good rewards-based obedience class yet? If not -- that should be a number one urgent priority. She hasn't earned the ability to NOT be kept closely managed and she is getting too much unsupervised freedom (also the pen incident would indicate this -- that was a very close shave for you and could have cost you thousands to resolve at the vets). Is she still under a year? She sounds like a normal if hyper puppy, or a very active adult -- but not abnormal behaviour if she hasn't had much training and doesn't have enough to keep her brain busy
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You will need to approach this through two prongs -- careful daily management as you (of course!) value keeping her alive, and good training with daily practice (a key responsibility of every dog owner -- a dog doesn't know to return on recall. Just knowing her name isn't recall; that requires a lot of training and work and is a critical command, as you can well understand!).
I'd invest in some baby gates and a puppy xpen with sides at least three feet tall, to manage her. If you are visiting, I'd bring the xpen or keep her on a lead the whole time and tethered to a belt loop or belt (do NOT use an extensa lead -- just get a 4-6 foot lead so she can move around a bit). Keep hall doors closed. If there are small kids opening doors, then stop that by putting a latch high up out of reach so that children need an adult to open doors for them and cannot let the dog out.. the exact same as would be done if you had a toddler. I always advise thinking of a dog as a toddler -- you simply cannot expect a toddler to mind itself, not wander out an open door, not to run into the street. A responsible adult must mind a very young child. The same for a dog in the house -- especially with a dog that is prone to running outside.
You need to be able to train her to sit immediately on command, to return, to wait. I'd train her to sit and wait before the front door every single time she is going out and wait to be released before you go out. Also, if she were in a puppy pen, she won't be able to bolt for doors when kids are around opening doors, or have access to something like a pen. If those items are kept off the floor and low tables, she lso won;t get to them (all these things are frustrating but I am afraid, a part of daily management for every one of us when we own dogs, just as you do not leave scissors, knives or matches lying around when you have small children in the house. You have to dog proof a house when you get a dog and change how things are done and what is left lying around).
All of us will likely experience the rare crisis of a carefully watched dog getting out, but this really must be the the rarest exception to the daily norm of careful guardianship. Cavaliers in particular have absolutely no road sense and are fearless -- it is is the very breed description used by breeders! -- so will happily run right in front of a car. That is why this breed in particular needs to be controlled and/or on a lead *at all times* without any possibility of sneaking out of a garden, or running out a door.
I'd recommend downloading Ian Dunbar's free book After You Get your Puppy (search for the link on the site as I have posted in a hundred times! :lol
and use that to start your training programme and for management ideas. Also there are many good suggestions on the training sites pinned at the top of the training forum, any of which should be helpful. Read the "if you only read one thing' post pinned there.
But you also need an organised class or your dog will not learn to respond when there are distractions around.
If she is slipping her harness, see a trainer about appropriately adjusting it or try a front clip harness (eg Easywalk or Sense-ible)where you can attach the lead to the front ring AND the collar at the same time for extra control and safety. I do this with my Lily.
She also sounds very active and bright -- and perhaps a dog without enough to do at home -- do you use active toys like stuffed Kongs? Treat balls? Does she get at least an hour of walks and active play with people --not on her own -- every day? A chance for regular supervised play with ther dogs?). If she is an adult dog with this level of energy I am afraid you will need to put the time in to working her brain as well as her body (my Jaspar is like this and has not tired even at age 6.5 -- they are not easy dogs when they are bright, very active and demanding. They can be wonderfully rewarding for the right owner but take a LOT of time and effort). This kind of dog needs at least an hour of active work -- agility, obedience, games -- daily. Also: consider whether she is the right dog for your family, a much harder decision, but if the choice is her staying alive or not because she is more active and needy of supervision and daily activity than can be given to her -- then surely that must be a consideration as you will end up losing her tragically rather than seeing her happily and safely rehomed. I know these are all big things to consider but from the sounds of it, the situation is fairly urgent and worrisome, and her survival is becoming dependent on taking some definitive action.