Not exactly sure about the info you are getting. Unless you will be working for a College, University, non-profit organization or government research facility, all of the H1B's are gone from last year which would qualify you for work this year. Second, you can only apply for an H1B on April 1st of the current year, but it won't be effective until October 1st of that year - so I don't quite understand the part about August 10, unless again you are working for one of the exempt categories I first listed.
There are only about 58,000 H1B's available for those with a bachelor's degree or equivalent and 20,000 for those with a masters. Since April 1 was on a Sunday last year, they received over 120,000 applications between April 2 and 3 to fill the quota. It ended up being a lottery in the end and all the extra applications were returned. So unless you are in the exempt category, then you actually have a different issue.
So assuming you are not in the exempt category your problem is that first you will have to hope that you will win an H1B this year as they will surely all be gone on April 1st. Second, your problem will actually be from the time you start with your new employer to the time an H1B kicks in. Here are the issues:
1. If you don't get an H1B for this year then you can only get a TN - and that's assuming your new job is still in the TN category.
2. If you do get an H1B, and then apply for a TN, your new TN will negate the H1B even though it's only to "bridge the gap". USCIS looks at "last action taken" as your status.
The timing on all of this is tricky. Since you can have multiple TN's, what you might want to do is see if you can get a second TN for your new company now. That will hold you from now until next year at this time. If you are able to get an H1B, then on October 1, your H1B will take over.
Also, were you planning going to Canada and come back to the US for H1B stamping purposes? If so, you don't need to if your H1B I-129 was sent in with a change of status request. If you are going just for a visit, then you will still come back as a TN until October 1.
This is only a discussion forum so you may need to get some legal advise from an immigration attorney because of your timing issues.