Yes, companies owe the full Delaware franchise tax each year they are in existence - even if they only existed for a small part of the year. So if you incorporate a company on December 31 of a given year, that company will owe the full Delaware franchise tax for that year.
The same is generally true for California franchise tax. However, newly qualified C-corporations subject only to California's minimum franchise tax are exempted from the franchise tax in their first taxable year, pursuant to Section 23153(f)(1) of the California Revenue and Taxation Code. Note that if your company starts doing business in California, it is subject to taxation in California even if it has not yet qualified to do business in California.
One final consideration is that there may be other considerations that would make it beneficial to incorporate prior to the year-end. For example, in recent years, there has been a partial exclusion of capital gains for qualifying C-corporation stock issued prior to the end of the year (read more). The deadline has been extended several times, but there is no guarantee that it will be extended in the future.
Taxes can be very complicated - unfortunately, we cannot provide anything beyond the basic general information here. If you have any further questions, please consult an attorney or accountant.