Universities with 10 month contracts design them so they do not have to pay new or leaving FTs for the summer months when they are not teaching.

The contract itself is not the issue, it is the residency permit. The university who 'sponsors' the residency permit is responsible for the FTs behavior and location during the duration of the permit. In the event that you acquire a new employer, the issue is: Who takes responsibility for you during the month or two prior to your arrival at the new University? The residency permit is supposed to conform to the dates on your contract, but I have seen provinces provide RPs that extend beyond the contract dates, bridging the summer months in case the university elects to extend your contract.

Your new university has a vested interest in you arriving to teach. Have their FAO dept contact your current FAO dept to arrange for a 30 day extension (which is common) from the end date of your current RP and/or the new university can date your new contract from August 1 with a contract amendment which specifies that August is an "unpaid orientation period", salary begins at the date of your first class.

Requiring you to fly home and start a new Z-visa process is inane. If your new university refuses to make the adjustments above, find another employer.