Their only child I could find in the extracted records (from the parish register) on the IGI at Familysearch.org is:
EDWD KARRIN Christening: 24 JUL 1822 Saint Mathews, Douglas, Isle Of Man. Father: WM KARRIN, Mother: JANE CALLISTER.
The Source is LDS film No. 0106718 [Baptisms, 1705-1848 Marriages, 1730-1753]

The other entries with William and Jane as the parents are Church members' submissions which may or may not be correct. Some of these entries have replaced genuine extracted entries, and some have been invented to fit someone's family tree.
One is a bapt for Jane CARREN Christening: 14 JUN 1818 Parish Of St George, Douglas, Isle Of Man, which is the same year you gave for Elizabeth, and because it is a church member's submission may be a misreading.

I think you need to look at these two parish register films (for St George & St Mathews) to see whether you can sort out the family before going to the IOM. It is best to find out as much as you can before leaving, to save time when there. It is also a good idea to join the IOM FHS if you are not already a member, to get help while there. See: www.iomfhs.im

The IGI has another member's submission for a marriage between WILLIAM CAREN & JANE CALLISTER on 28 OCT 1817 Braddan. Again, this could replace a genuine entry, and it would be a good idea to look at the parish register film.
Marriages 1713-1817 LDS Film No.0106709
Marriages 1817-1844 LDS Film No.0106710
[This is annoying, apparently splitting 1817 - perhaps the later one 0106710 would be the better guess with the date given as 28 Oct?]

The other (third) film you need to see, for St George bapts in 1817 is LDS film No.0106717 [Baptisms 1781-1853 Marriages, 1786-1896 Burials, 1790-1892].
"Douglas is a town partly in Kirk-Braddan and partly in Kirk-Onchan parishes. The churches lie within the boundaries of Kirk-Braddan parish."

LDS films are available through local Family History Centres, worldwide. Look in the local phone book under "Church of Jesus Christ of Latter-Day Saints" to get their address and non-member opening hours. They charge a small fee to order the film to be delivered to them and it is yours for a month to read and photocopy on their film readers. They ring you when it comes in, and you book a reader for when it suits you. You can usually order several films at a time.

Sue