Edit distance when substitutions cost more

An OCR pipeline reads scanned trade confirmations and must correct garbled fields against a clean reference. Not all mistakes are equal: a dropped or spurious character (insert or delete) costs $1$, but a substitution costs $2$, because a swapped character usually signals a worse mis-read and you want the aligner to prefer explaining errors as insert-plus-delete when that is cheaper.

a = "abc", b = "adc", ins = del = 1, sub = 2
-> 2           # substitute b->d costs 2, or delete b + insert d also costs 2

Compute the minimum-cost edit distance with general per-operation costs. Target $O(|a|\cdot|b|)$ time.