"I think this is mine."
It is still grammatically correct, but is useless socially. So I am afraid the answer is no.
In the case you need to be extra polite, e.g. talking to your business customer, you actually need to say a largely different sentence.
I do not see any grammatical difference between だ and です。They are different in the level of politeness but that is not a grammatical issue to me.
I would love to learn it if you can explain more.
Go to the section: 「です」 is NOT the same as 「だ」 Although they normally can do the same thing, there are some specific situations where you can only use one of them, it's worthy to have a look at it ;)
Thank you for the link to Tae Kim's page. I have actually come across the link before and bookmarked it for my future reference, but haven't read it yet till now.
I read the part where だ<>です is explained. However I found the explanation there a bit out of way. Also the explanation for how a question is formed out of そうだ is barely convincing. I feel that caution is needed when using this website.
Just to make it clear, there is no disagreement here that ~ですと思います is not a correct form to use, which should rather be ~だと思います。The discussion is merely about the reason behind.
です is widely agreed to be the polite form of だ. Both of them are of the same type (助動詞) and in its ending form (which we can put a fullstop next to it). Thus there is no reason, at least technically, why they don't fit in the same position of a certain sentence structure. All these features can be looked up from a reliable dictionary.
when used for referring to content, the requirement is "文や句をそのまま受けて" meaning receiving a sentence or phrase as it is. (Usually we generalize it as the ending form, but according to some native discussions, it is only largely, but not exclusively, true.
Among all the resources that tells us which form is required without giving a reason, I managed to find a book in Google that gives a hint. It is calledはじめての授業キット: 日本語を教えたい人のための First Teaching Kit: For People Who Want To Teach Japanese.
In page 173 it says
There may be people wondering why it is 「~ます」at the end of the sentence but the plain form comes in the middle. We can tell them that because politeness is determined by the expression at the end of the sentence, even when we want to say it politely, the 「<sub>」in「</sub>と思います」is always in plain form.
which seems to be a sensible way of explanation to me.
Normally you aren't going to see or useです in the middle of a sentence, only at the end in polite Japanese. Just use だ in any "nested" sentences like the middle part of わたしは。。。と思います unless it's a direct quotation.
In this case だ corresponds to "is", which you wouldn't leave out in English, either. If you were ending the phrase with a verb or an い-adjective, you wouldn't need だ.