Oddly swear by promises to pay for grave (7)
I believe the answer is:
serious
'for grave' is the definition.
(similar in meaning)
'oddly swear by promises to pay' is the wordplay.
'oddly' indicates alternate letters (letters in odd-numbered positions).
'by' is a charade indicator (letters next to each other).
'promises to pay' becomes 'IOU's' (promises to pay a debt).
The alternating letters of 'swear' are 'ser'.
'ser'+'ious'='SERIOUS'
(Other definitions for serious that I've seen before include "Earnest, sincere" , "Important -- grave" , "Earnest - important" , "Earnest; vital" , "Not light or frivolous" .)