It looks like aeternē was rarely used in Classical Latin. The PHI corpus shows only three matches, two of which are in grammatical discussions explaining that aeternum means aeternē.
The third is in Cicero De Natura Deorum 1.109.6:
quo modo enim probas continenter imagines ferri, aut si continenter, quo modo aeterne?
The Lewis and Short entry for aeternus has an entry for adverbial expressions that mentions aeternum and aeternō, and also the phrase in aeternum:
Adv. phrases.
- in aeter-num.
A Lit., forever, everlastingly: et vivat
in aeternum, Vulg. Gen. 3, 22: hoc nomen mihi est in aeternum, ib.
Exod. 3, 15: Dominus in aeternum permanet, ib. Psa. 9, 8: vivet in
aeternum, ib. Joan. 6, 52: Tu es sacerdos in aeternum, ib. Heb. 5, 6:
non habebit remissionem in aeternum, ib. Marc. 3, 29.—
B Meton., of
indef. long time, forever, always: urbs in aeternum condita, Liv. 4,
4: leges in aeternum latae, id. 34, 6: (proverbia) durant in aeternum,
Quint. 5, 11, 41: delatores non in praesens tantum, sed in aeternum
repressisti, Plin. Pan. 35: (famulos) possidebitis in aeternum, Vulg.
Lev. 25, 46: (servus) serviet tibi usque in aeternum, ib. Deut. 15,
17: ut sceleris memoria maneat in aeternum, Lact. 1, 11.—
- aeternum.
A Lit., forever: sedet aeternumque sedebit Infelix Theseus, Verg. A.
6, 617: ut aeternum illum reciperes, Vulg. Phil. 15 (prob. here an
adv.).—
B Meton., of indef. long time, forever, always: serviet
aeternum, Hor. Ep. 1, 10, 41.—
C Of what is continually repeated,
constantly, again and again (as in colloq. Engl., everlastingly,
eternally): glaebaque versis Aeternum frangenda bidentibus, Verg. G.
2, 400: ingens janitor Aeternum latrans (of Cerberus), id. A. 6, 401.—
- aeternō, meton., of indef. long time, forever, perpetually: viret
aeterno hunc fontem igneum contegens fraxinus, Plin. 2, 107, 111, §
240: BVSTA TVTA AETERNO MANEANT, Inscr. Orell. 4517.
I don't have enough reading experience in Latin to have a sense for the differences, but a more experienced Latinist, Kingshorsey, made the following observation in a Reddit post:
The most common adverbial form is in aeternum, followed by aeternum and aeterno.
(2019/1/23 reply to "Translation request: Licet amētis permansimus, nunc, et aeternum.")