I THINK that I might have found the right family after studying census and cross-referencing against some online family trees. This was my journey of research.
James Dewar appears to have married an Agnes Edmond/Edmund ca.1895 - in 1900 they are living in Bloomfield, Essex, New Jersey with Agnes' widowed mother.
Mary Edmond 58 wid. born Scotland
James Dewar 29 (shown born Aug.1870 Scotland) Night Watchman Immig.1889
Agnes Dewar 26 (born May 1874 Scotland) Immig.1890
Agnes Dewar born April 1898 New Jersey
Mary Dewar born October 1895 New Jersey
William Dewar born Sept.1899
New Jersey births seems somewhat odd and you can't view original document on Ancestry but there are 2 births showing James/Agnes Edmond as parents - one for a male transcribed as Duer born 14th September 1899 (must be William) and one for a female transcribed as Dues born 18th April 1897 (think this must be Agnes so not born 1898 as shown on 1900 census). What threw me a little here is that the birth entries state James was born in America, Agnes in Scotland).
Bear with me.
Next I looked at 1910 census and found in Hudson New Jersey:
Annie Dewar 70 Wid. Immig.1885
James Dewar son 36 Widr. Stone Cutter Immig.1885
Benjamin Dewar son 34 Widr. Stone Cutter Immig.1885
Mary Dewar grand-dau. 15 born New Jersey
I found the Dewar's actually arriving in the US in 1886 - Ann Dewar 46 arrived 19/5/1886 on State of Nebraska and appears to have arrived alone. Two months later on 21/7/1886 on ship State of Georgia arrives James 13 plus 3 younger siblings - Chr. 10, Benj. 8 and Jessie 6.
The Scottish naming pattern has eldest son named after paternal grandfather so as James' first son was named William I was hoping to find a William Dewar married to an Ann/Annie and believe this is the family in 1881 in Hutchesontown, Glasgow, Lanarkshire:
William Dewar 48 Mason born Cupar, Fifeshire
Ann McMillan Dewar 41 born Bellshill, Lanarkshire
Grace Dewar dau. 22 born Strathbungo, Renfrewshire
Thos. McMillan Dewar 15 ditto
Margaret Dewar 15 (think this should be 13) ditto
James Dewar 10 America
Christina Dewar 6 America
Benjamin Dick Dewar 3 born Govanhill, Lanarkshire
Jessie McMillan Russell Dewar 1 ditto
No sign of William, Ann and the 4 younger children in Scotland 1891 - can't see an obvious death for William 1881-1886 on Scotlands People site so I'm guessing William went to US first (haven't found his arrival), with wife Ann and the 4 children arriving in 1886.
So, although James Dewar was 'Scottish' he was actually born in the US although the year varies from document to document i.e. 1881 census shows bc.1871, immigration shows bc.1873, no census in US 1890, 1900 census states born Aug.1870, and 1910 census states bc.1874. So, whilst his three youngest children were put in Foster Home after death of his wife Agnes, he appears to have kept his elder daughter Mary b.1895 with him.
James appears to have been born in August 1870-1873 in America (as was his sister Christina ca.1875) before family returned to Scotland. From one of the census I looked at his mother states immigration 1868 and I'm assuming that she and William went to the US then where they stayed for around 5/6 years and then returned to Scotland. This was for around 10 years as back in the US in 1886. So, Aunt Tini is sister Christina, Uncle Ben is brother Benjamin, and I'm wondering if the remembered 'Betty' was in fact his youngest sister Jessie??
James' parents William Dewar and Ann McMillan married in 1855 Cathcart (Glasgow), Lanarkshire. If you purchase some credits for Scotlands People you will be able to view the marriage certificate which will show respective parents for each of them and take you back yet another generation.
Phew - hope I've deciphered all my scribbled notes correctly but this certainly does look as if this is your family.
Annette