Thanks for the question.
In direct answer to your question : The sideways movement of the eye is called nystagmus and you may well see it in cases of vestibular syndrome, inner ear infections, and also some neurological conditions such as brain tumours. This is just part of these diseases but hopefully may be temporary.
While I cannot diagnose this over the internet I have seen many dogs in my professional life which had incoordination, a head tilt and nystagmus, while some very due to serious diseases many cases were not that serious and could be treated. I can list all the possible causes of the symptoms you describe based on my previous experience, these would be :
1. Some kind of middle ear infection.
2. Consider idiopathic vestibular syndrome : I have seen this frequently in older and middle aged dogs. There is no real treatment and it will resolve on its own in many cases.
Here is an link to an excellent article which covers the subject in depth : LINK
3. Sometimes it can be some form of neurological disease such as a brain tumour rather than a middle ear problem. While rare this can happen .
Quite frankly there is nothing you can do on your own on a home remedy basis so all you can do is to have your vet check your dog over as you did and to continue to work with your vet on this one.
If I have not covered your question fully enough or you would like to ask more I will be online for the next hour or so and I will be at your disposal.Scott Nimmo BVMS MRCVS