The Cleveland FHS have transcribed the MIs for Lythe St Oswalds and a copy can be obtained from their web site for £2.95.
https://clevelandfhs.org.ukProbate records at the Borthwick for people called LINTON near Whitby:-
Ann LINTON April 1846 vol 213B f. of East Row P. Whitby Admon £450
Elizabeth LINTON June 1836 vol 193 f. of East Row P. Whitby Admon £800
Henry LINTON Sep. 1761 vol. 105 f.174 of Lyth Will
Henry LINTON Dec. 1814 vol 158 f.547 of P. Whitby Will £7500 - DDR (Death Duty Register) says the executor was John LINTON of East Row, Whitby.
John LINTON July 1770 vol. 114 f.148 of Ugthorpe P. Lythe, Gent. Will
John LINTON June 1777 vol.121 f. of Ugthorpe P. Lyth Admon
John LINTON Nov. 1783 vol.127 f.501 of Ellerby P. Lyth Will
John LINTON Dec. 1828 vol.178 f.390 the Elder of East Row P. Whitby Will £2000.
John LINTON July 1839 vol.200 f.81 of East Row P. Whitby Will. - DDR adds his abode was East Row Newholm and Executor was Mary WATSON of same place.
John LINTON July 1847 vol.217 f.41 of Ugthorpe P. Lyth Will £450. DDR adds Executor was Matthew LINTON of same place.
Matthew LINTON 28 Nov 1711 of Lyth Admon.
Matthew LINTON June 1777 vol.121 f. of Lyth Admon
Matthew LINTON June 1834 vol.189 f. 64 of Ugthorpe P. Lythe Will £200
Before spending money on any of the Wills or Admons (Administrations) check to see if abstracts of the Wills can be found in Dave KING's site -
http://www.davekinggenealogy.co.uk. Some of them are.
Note that Admons usually contain little information; just a note of who the Executor was. Though I do have one which listed all the children and said that three of them had emigrated to America.
If you can get to York then you can download images of the Probate records to a USB stick for free. Otherwise order from -
https://www.york.ac.uk/borthwick/remote-services/copying/order-form/ PS - and the name LINTON is mentioned a couple of times on this web page about Wills -
http://www.hoggarth.co.uk/wills.htm