I'm not sure why you've had no replies,but let me be the first to say welcome to Rootschat.
It's almost traditional that you get lots of questions instead of an answer, at least initially! So here goes ---
Where did you get the information that you have so far on Solomon? and where did you get the middle name, as I can't immediately see that on any records.
You give his death year as 1856; there was only one death for Solomon Dawson registered in that year, and that was in Liverpool - is this your man? But the GRO gives his (reported) age at death as 40 ---
There are 2 Solomon Dawsons dob 1808 in the 1851 census. One was an army pensioner, born in Cornwall, lodging with his wife Mary and 3 daughters in another household in Liverpool. The other was born and was still living in Pilkington, again with a wife Mary and several children, and was a cotton weaver. This Pilkington man appears in the 1861 census. Is either of these men yours?
If you can give us more info (and tell us where you go the info from), we can try to work from there.