Is the "ng" sound often pronounced simultaneously with the "n" sound?

Firstly, native English speakers do not add an extra n after the /ŋ/ in the word singer, though some do have the same issue as you with having difficulty producing /ŋ/ in the middle of words like singer and singing.

The "ng" sound in medial and final word position can be challenging for native Russian speakers. What's happening is that /ŋ/ is a continuant sound, and if you release the tongue from the sealed position for the /ŋ/ while you are still producing airflow, you will make a /g/ or /k/ sound because the release of the built-up air produces the sound.

The way to avoid this problem is to begin to shape the vowel for the "-er" before releasing the /ŋ/ tongue position, and to release the tongue gently into the vowel. If no air is allowed to build up, the plosive/stop g sound will not be produced.

You can hear the pronunciation of singer in this video: https://youtu.be/c-3HtmE5muY and get more information about the /ŋ/ phoneme in American English in this video: https://youtu.be/-DZ5GICTHVU

I will be making a more specific video to help with words with "-nger" and "-inging" in the future, as I have heard from many Russian and other non-native speakers that these are particularly difficult to make without the "g" sound intruding. I'll add it to this post as an edit when it's complete.