Why has "sware" become "swore", "bare" "bore", etc?

Tear and bear were in Old English (OE) both strong verbs of “class 4”. The vowel pattern for verbs of this class is "e" - infinitive, ‧ "æ" - past singular, "ǣ" - past plural, "o" - participle (Wiktionary). This class seems to mostly be composed of verbs that had a liquid after the ablauting vowel; another example is “steal” < OE stelan, with /l/ rather than /r/.

The regular development of these vowels in Middle English would be /ɛː/, /a/, /ɛː/, /ɔː/ respectively, which indeed seem to have been in use ("Middle English Morphology" by Jerzy Wełna, in English Historical Linguistics, Volume 1, edited by Alexander Bergs and Laurel J. Brinton).

The Oxford English Dictionary (OED) says the past-tense forms in -are are attested starting from Middle English. At this time period, this spelling would have represented something like /aːrə/ or later on /aːr/. What seems to have happened is that, at some point in Middle English, past-tense forms developed that show the continuation of the vowel from the Old English singular past tense forms, but that have an -e suffix that lengthened the vowel and caused voicing of a fricative consonant. See for example drove < OE drāf and gave < OE gæf/geaf. The OED entry on "bear" describes this process a bit and specifies that it started in the north:

early Middle English bar, beren, afterwards by levelling of singular and plural, in south ber, beren, beeren, in north bar(e, baren, bare, which became the literary form.

Analogical shift of other -ear verbs to this class

A type of analogy caused some verbs that were not originally strong class 4, such as swear (originally strong class 6 with ablaut pattern e - ō - ō - o) and wear (originally a weak verb) to change conjugation patterns to join tear and bear. (Swear not only had the same vowel as tear and bear in the present tense, but also in the usual form of its past participle, which may have facilitated this shift.) This is why sware and ware exist.

Paradigmatic leveling of "o" from the past participle of strong class 4 verbs

The use of “o” in the past tense in Modern English is apparently due to a another type of analogy: "leveling" within the inflectional paradigm in favor of the vowel of the past participle. The OED entry on “tear” says:

The Old English past tense tær ( < tar) survived as tare to 17th cent., when it gave place in standard English to tore, with o from past participle toren, torn: compare bore, swore. A weak past tense and participle terede, tered, found in 15th cent., are still dialectal, along with a mixed form tored, tord.

I don't know why the vowel of the past-participle won out. I would guess the use of "o" may have also been influenced somewhat by the use of “long o” (Middle English /ɔː/) in the past tense forms of some other strong verbs, such as class 1 “drive~drove”. Avoidance of homophony with the present tense might also have been a factor, although that's very unclear (there are a number of verbs in Modern English that exist well enough with homophonous present and past-tense forms, such as put and cut).

This sort of analogy is definitely the source of tore, bore, wore. With swore, the OED says it could be a continuation of Old English swór, although that would be slightly irregular phonetically since we'd generally expect the vowel to be /ʊə/ in that case ("swoor" or "swoore"). However, vowel lowering before "r" is a well-attested sporadic sound change in English: it happens in the infinitive/present-tense forms of all of these -ear verbs, and it also seems to have occured in the words whore, floor, and probably door. So it seems quite possible to see swore as a continuation of swór, but analogy doubtless played a role as well.

The power of these two forms of analogy was apparently so strong that we even see unetymological “o” appear in the past participle and past tense of some verbs that don't end in "r": the words speak, tread, weave, which come from Old English strong class 5 verbs (Old English ablaut pattern: "e" - infinitive, ‧ "æ" - past singular, "ǣ" - past plural, "e" - participle). The OED says about “weave”:

In the 14th and 15th cents. the form of the past participle became assimilated to that of the past participles of strong verbs with root ending in a liquid (e.g. steal, stolen), and, as in most verbs of that class, the o of the past participle was extended to the past tense both singular and plural. The weak inflection has been occasionally used in all periods from the 14th cent. onwards, but has never become general.

As you may know, "spake" is an archaic Modern-English past-tense form of "speak".

A full explanation would take more than this site allows, and would no doubt try the patience of many. To give just a flavour of what might be involved, here’s part of the the OED’s etymological note on swear:

The conjugation of this verb has been influenced from early times by that of bear (Old English beran ). The regular past tense swore (Old English swór ) has never ceased to be extensively current, but from the 15th to the 17th cent. sware , formed on the analogy of bare (Old English bær , bǽron ), was widespread; swar occurs as early as the first text of Layamon; suar(e) is the prevailing form in the Cotton MS. of Cursor Mundi; sware and swore are both used in Malory's Morte Darthur; sware is the only form in the Bible of 1611 (exc. in the Apocrypha), but is rare in the 1st Folio of Shakespeare. In the 14th and 15th cent. a by-form swere occurs, after bere .