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Unless the sites have exactly the same content or you interlink extensively between them there is no problem at all. The myth of duplicate content is greatly over-rated.
The extensive interlinking can be a problem. A few one way links won't hurt or add to your SER.
Look at what you wrote in your post. Wanting to brand the different segments of your business in a different manner is entirely logical and reasonable.
It also normal for the older site to have more back links than the newer sites. Build the backlinks and add more content to the site(s) you want to promote and over time their rating will rise.
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