As far as I know, the historical reason for the spelling of of with <f> is simply because at one point it was pronounced with /f/, as the etymology and relation to off suggest. I don't think Welsh influenced English spelling in this area. Actually, the Welsh spelling convention appears to be derived from English influence, but most likely not the influence of this particular pair of words. The use of <f> to represent [v] in English texts was more common in the past, and occurred in other words.
As you've found, in Old English the letter <f> was used to represent the sound [v] (a voiced labiodental fricative) between vowels. This occurred as the result of a process of voicing earlier voiceless [f] in this kind of context; also, past a certain point, the letter <f> was used in Old English texts to represent the [v] sound that developed from earlier [b] between vowels (A Grammar of Old English, Volume 1: Phonology, by Richard M. Hogg, 2011: 2.58). Old English [v] is standardly analyzed as a conditioned allophone of /f/, as the distribution of [f] vs. [v] is generally predictable (there are a few not entirely straightforward cases like the use of voiceless [f] in the suffix -ful/-full). The digraph <ff> was used in Old English for the sound [ff], a long voiceless labiodental fricative, but this was fairly rare (Hogg says it occurred mainly in proper names like Wuffa, onomatopeia like pyffan 'breathe out', and words that originated as loans from Latin like offrian 'offer').
The use of the letter <f> to represent /v/ in Welsh is thought to be due to influence from English spelling conventions according to Lectures on Welsh Philology, by Sir John Rhys, 1879 (Lecture V). In older forms of Welsh, <v/u> could also be used to represent this sound (<v> and <u> were not originally distinct letters), and apparently <w> was also sometimes used, although this may represent an alternative pronunciation with the phoneme /w/ (such variation between pronunciations with /v/ and /w/ apparently exists today for some words in modern Welsh dialects, such as cawod/cafod 'shower', which David Willis says indicates that Welsh /v/ probably was pronounced [β] in Middle Welsh (Old and Middle Welsh PDF)).
A Welsh Grammar, Historical and Comparative, by J. Morris Jones, 1913, says in the chapter "Phonology" that
In Late Ml. W. the sound f was written medially u or v and fu; finally it was represented by f regularly (the few exceptions which occur, e.g. in W.M., being due to mechanical copying). Thus, IL.A., vy 2 ≡ fy ‘my’; llauur 3 ≡ llafur ‘labour’; kyfuoethawc 55, Mn. W. cyfoethog ‘rich’; gyntaf 3 ‘first’, dywedaf 3 ‘I say’, ef 3 ‘he’, etc. u and v continued to be used medially for f during the Early Mn. period; but G.R. has f everywhere, and was followed by Dr. M. in the 1588 Bible, which fixed the Late Mn. orthography.
According to Rhys, there is no relevant sound change in the history of Welsh that would really help explain the use of the letter <f> for the voiced sound [v], and so, he says
we turn to English for our answer. Now, Anglo-Saxon words like heafod 'head,' heofon 'heaven,' næfre 'never,' had their f pronounced v, and sometimes it was also written u or v, and not f. Further, we are told by Mr. Ellis (Early Eng. Pro., ii 572) that, in English manuscripts of the 13th century and later, ff was used for the sound of ph, and he gives extracts from Orrmin dating from the end of the 12th century. From the latter it is clear that he observed the same sort of distinction between f and ff as we do in Welsh: his f between vowels was mostly v, while his ff had, of course, the value of ph. Neither is it altogether irrelevant that the pronunciation of f as v was most prevalent in the West of England, and that it survives extensively in Somerset and Devon. Salesbury noticed it in his time ; his words are : "I my selfe haue heard Englysh men in some countries of England sound f, euen as we sound it in Welsh. For I haue marked their maner of pronounciation, and speciallye in soundyng these woordes: voure, viue, disvigure, vish, vox: where they would say, foure, fiue, disfigure, fysh, fox" &c. (Ellis's Early Eng. Pro., iii 752). In the Black Book, of the 12th century, and in the Book of Aneurin, partly of the 13th century, f initial did duty for the sound of ph and between vowels for either ph or v, but when a little more consistency became the rule, ph was usually confined to the mutation of p, which we still so write, while the same sound was elsewhere written ff, not excepting when it happened to begin a word. How early ff began to be used as an initial in Welsh I cannot say, but it appears in that capacity in the Book of Taliessin of the 14th century. That the Welsh should have so used it at all is not surprising, seeing that they had before them the analogous case of ll, as well as probably the same use of ff in English, which would explain how it came to be sometimes regarded as a mere equivalent for a capital F. Later, we find Salesbury treating R and rr in the same way ; and perhaps in some of the proper names written with ff, such as Ffoulkes, Ffrench, and the like, the digraph is neither Welsh nor modern. It is worth adding that English manuscripts of the 13th and the 14th century show instances of ss, initial as well as medial, for sh, and that Welsh dd has also been traced back into the 14th century.