What's a proper title when I started a post-doc position but not received my PhD degree yet? I finished my Ph.D. research in computer science a few months ago and started writing my dissertation. Meanwhile, I applied for a postdoc position before my Ph.D. fund could run out. So, I found and started a postdoc position around a month ago in another University in Germany, and I'm going to have my disputation in 3-4 months from now (that's the usual waiting period in CS in Germany).

However, I'm not sure what title I should use regarding my position in the new institute until I do my Ph.D. disputation, especially in email correspondences and when I introduce myself or other similar situations?

Of course, I do not use the "Dr." title until I do the disputation, but should I declare myself still as a "Ph.D. researcher" or a "Post-doctoral researcher" or something else regarding my position here?

BTW, my contract just says a research employee.
I would use research fellow as your title in English and in German use whatever your contract says or research fellow as well. That leaves out the slightly awkward postdoctoral part. 

In general, unlike a doctorate, using the title 'postdoc' is not something that people in academia worry a lot about (and outside academia many people will have no idea what it means). So using postdoc to describe your role wouldn't be outrageous even though linguistically it seems wrong because of the implied doctorate.