'Shrammed', although old, seems to be a more recent variant on 'shrimped', meaning shrivelled or huddled up with cold. The earliest recorded use of 'shrimped' dates back to the 1630s and it was written by the son of a clergyman in Suffolk...so not really very West Country!
'Shrammed's' earliest appearance - in Francis Grose's Provincial Glossary - was a century and a half later, so all we can say for sure from the title is that it was a dialect word but not necessarily where it originated.
Of course, by the time that 'shrammed' version appeared, it may very well have been in Wiltshire/Hampshire.
As a matter of interest, here is a quote from an edition of The Daily Telegraph published in the 1860s "...being shrammed with cold, as they say in Wiltshire." So, after a couple of centuries, it would appear that it had become West Countryish even if it did not really originate there!