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Your videos are actually pretty useful as they demonstrate two important facets of singing. The one where you sing over the woman's voice is more confident and noticeably more in tune. This is because you have a reference pitch to follow and you can mimic along with the vocalist.
This is sometimes called 'shadow singing'. It happens a lot in amateur choirs when a singer who can perform very well with the group falters on their own, What you hear is a tiny, almost undetectable delay between the lead voice and yours as you hesitate an instant as you follow along. Comparing that to your acapella we can see your weaknesses emerge which is useful for analysis.
Singing without accompaniment is far more difficult and no other vocal form will test the singer's skills and abilities more. It is here that your shadow singing becomes glaringly obvious. Every aspect of your performance is being affected and you basically fall apart.
Breathing, breath support, phrasing, timing, intonation, tonal center, everything has gone to hell in a hand basket. You have no reference points to keep you on track. But do not be discouraged!
This is remarkably common and almost always correctable. You need a trained vocal coach, and right away. We cannot give any good advice based solely on a video any more than a student can learn effectively from videos.
On the plus side you have a nice, solid voice. You are likely a baritone, but that cannot be accurately determined here either. But get to a teacher and get to work.
You can learn to sing.
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