Learn English – Etymology of “evangel”

It is indeed a euphemistic or 'minced oath' version of "God love me" - with the verb, here, in the optative mood ("may God love me" or "as I hope God loves me") to invoke the Deity as guarantor of the speaker's sincerity. An analogue is "corblimey" < "God blame me", that is, May God impute sin to me (if I'm not speaking the truth).

• ADDED, 3/14/17: An anonymous reviewer suggests that blimey derives from blind me. That seems equally likely, and appears to be the sense in which the word was understood by late 19th-century observers.

It's 'cockney' in the broad sense of 'colloquial lower-class British', but it's not restricted to the area within earshot of Bow Bells and it's not rhyming slang.

In short, your only option seems to be exegese (or possibly exegeze, but probably not), if you must have a verb that would be acceptable to the OED.

But it should be noted that, for what is probably the majority of words on -sis, no verb is ever used; perhaps recasting your sentence would be a better idea, e.g. she interpreted the passage. Using the Greek infinitive exegeisthai is a bad idea that will send classicists off screaming.

The only traditional way of forming a verb based on a Greek word on -sis that I am aware of is -se:

Parenthesis => parenthese

Diagnosis => diagnose

Paralysis => paralyse

Basis/-se => base

Metamorphosis => metamorphose

Occasionally, modern forms on -sise are created, as in those based on -thesis (hypothesise, synthesise); I can't think of any other example. Those few Greek words on -sis whose noun-equivalents already end on -se in English also use -se for the verb, as in phrase and base.

For forms on -ise and -yse, you may encounter -ize and -yze in America, respectively, and possibly sometimes elsewhere too, so exegeze may be considered a valid alternative depending on your local tradition. However, this strictly has nothing to do with the formation of an English word based on Greek, but rather on regional variations within English. Lastly, I do not believe Americans normally do this with words on -ese/-eze (they stick with -ese: only after -i- and -y- is this z commonly used), but that's your call.

Your examples ostracize and baptize are different, because they have somehow retained or regained the z that was originally there in Greek (ostrakizein, baptizein), as opposed to in exegeisthai and most other verbs. The same applies to apologize/apologizein. This confusion is the reason (most?) British publishers and style books use -ise/-yse for verbs based on Greek nouns on -sis (Oxford and Cambridge alike, I think), like analyse; but they do often do use -ize for other Greek verbs, like baptize, and for words taken from non-Greek stems, like immunize, realize, colonize (either Oxford or Cambridge—I forgot). They usually do not, however, use -yze where Greek had no -uzein (so practically never). I know, it is a bit tiresome having to remember the exact origin of such words—I am usually too lazy to do so, I must confess.

The -i- in *exegise/exegize is not really defensible, because that is normally not done with words derived from -esis, but only with those derived from -isis. The -ize in energize comes from Greek -(e)izein, as energy comes from energ(e)ia; the -i- in apologize from Greek -izein as in apology, from apologia. The -y in English represents -(e)ia in most Greek (and Latin) words, and hence nouns on -y in English have verbs on -ise/ize.