Greetings Kevin, (and All)
Sistering will add strenth to the joist. I would add a piece of ripped, plywood between the 2 X 6 and the 2 X 10 and use an adheasive as well as screws.
If you plan on using the 2 x 10s, you need to make sure that it is properly supported. You are adding quite a bit of weight by doing this.
I don't know what your situation is so, I would recomend you talk with your local Building Inspector to find out what he/she will accept.
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