I am so sorry that you were having trouble. Vertigo or having dizziness problems can be a real disruption to a productive, enjoyable day for sure. Life is no fun with Vertigo.
I don't know about Coconut oil being stronger than other oils, although it very well could be I suppose, due to it's apparent small molecular structure (i.e. it is said to absorb better than some... or so I've read). It is also quite possible as MrCuddly has stated, that you very well might have an allergy to Coconut. It IS known to be an allergen with some people. That would be a crying shame however as Coconut oil, for me, has been by far the best for teeth whitening =)
With all this said, I'm not sure I would discount oil pulling just yet. First of all there is a likely possibility that your Vertigo and dry heaving was due to the Margarita ;), especially sense it happened after already OP-ing for 6 days. The fact that your Vertigo quit a day or so after you quit the Oil Pulling, might only mean that you coincidentally also gave your body enough time to heal from the alcohol trouble.
If you really don't think that's it, and you are concerned of a possible allergy to Coconut, then you might want to consider another oil. SUNflower oil (as opposed to Safflower) has always been listed as one of the age old suggestions for oil pulling, maybe you could try that one out(?). It is fairly flavorless too, which would be an added bonus here. For me, I have noticed that ANY oil works - Oh and that business of needing to do it on an empty stomach doesn't make a difference to this bod' either. I Oil Pull whenever I feel like it and with whatever oil I choose.... and I have GREAT results.
In reference to your Vertigo trouble & why I suggest you might want to consider continuing it, I will very briefly explain my personal experience with oil pulling.
I've been doing it on-and-off for about the last year and a half, and just about every time I start it up again after an absence from it (even if just for a week sometimes).... I can expect some sort of "negative" response. Usually it's only after the 1st round or 2, but sometimes it has continued a bit longer than that. Those responses always go away though as I regularly keep up with the Oil Pulling.
It may help you to feel better by knowing that I have, once before, experienced a large loss of hearing in both of my ears for a day (no lie I was a little freaked) which, by the way, Vertigo/dizziness typically happens to also be middle-ear related. I have also gotten very achy & flu like before as well as dealt with Salivary Gland Stones in my mouth numerous times - But ALL of this disappears as I continue to Oil Pull.
I suspect, for several reasons, that oil pulling is largely successful because it is lubricating our bodies - inside to the outside. When this fresh oiling of our body's system first begins to happen, I can only guess that old stagnant junk (virus, bacteria, candida, etc.) are being moved through our lymph system... like a car oiling it's engine and allowing parts to move smoothly and crud to flush thru... or like a trash truck moving our trash out of the neighborhood to a waste disposal sight after we Spring Clean. This, of course, would cause what is sometimes referred to as a "die-off response" - Junk that was hiding, now is shoved into our active trash disposal system so that it can be carried away forever.
When I get these uncomfortable responses after Oil Pulling, I make sure to listen to my body, DRINK A LOT OF WATER (not coffee, soda, etc. but WATER) so I can help flush out whatever has been stirred up, and then I will usually Oil Pull AGAIN after 6 - 12 hrs. The 2nd Oil Pulling and the water have been noticeably huge for me in speeding along the flush and feeling better. I highly suggest it.
So, I encourage you to considering continuing with the Oil Pulling but listening to your own body 1st and foremost. Don't mistake Die-Off as a "bad" response, however, don't discount that you could have an allergy to the oil as well. You are smart in stating that you should start with less time of swishing in the beginning. I would still suggest doing it twice a day at first, tho' for less time per session. It will allow your body time to process what it needs to at a more comfortable pace.