While I wish it was otherwise, there is virtually no way that "with" was intended in the case in Genesis 3:16. That sense is only when using it to mean how or by some means. It's not used in the cooperative sense:
Reference
Hebrew Phrase
Transliteration
Literal Meaning
Context / Relationship
Exod 14:25
בָּרִכבוֹ וּבְפָרָשָׁיו
bārikhbô û·bĕfārāšāyw
“with his chariots and horsemen”
Military accompaniment / equipment
Judg 5:15
וְשָׂרֵי יִשָּׂשכָר בְּדְבוֹרָה
wĕśārê yiśśāḵār bĕ·Dĕvōrāh
“the princes of Issachar were with Deborah”
Alliance / cooperation in battle
2 Sam 6:12
בְּשִׂמְחָה
bĕ·śimḥāh
“with joy”
Manner of celebration
Ps 33:3
בִּתְרוּעָה
bi·terûʿāh
“with a shout of joy”
Manner / instrument (musical shout)
Ps 44:6
בְּחַרְבִּי
bĕ·ḥarbî
“with my sword”
Instrument used in action
Ps 89:26
אַף־יָדִי תִּכּוֹן עִמּוֹ… אַף־זְרוֹעִי תְאַמְּצֶנּוּ
(parallel use)
“My hand shall be established with him”
Cooperative strengthening (parallel to ʿim)
Prov 3:19
יְהוָה בְּחָכְמָה יָסַד אָרֶץ
bĕ·ḥokmāh
“The Lord founded the earth with wisdom”
Means / instrumentality
Isa 11:4
בְּשֵׁבֶט פִּיו
bĕ·šēḇeṭ pîw
“with the rod of his mouth”
Instrumental metaphor (speech as weapon)
This present use is a dominative sense:
Reference
Hebrew Phrase
Literal Transliteration
English Meaning
Context
Gen 3:16
יִמְשָׁל בָּךְ
yimšāl bākh
“he shall rule over you”
Husband’s dominion post-Fall
Gen 4:7
תִּמְשָׁל בּוֹ
timšāl bō
“you must rule over it”
Cain urged to rule over sin
Gen 37:8
מָשֹׁל תִּמְשָׁל בָּנוּ
māšōl timšāl bānû
“will you indeed rule over us?”
Joseph’s brothers protest his dreams
Ps 8:7 (8:6 Eng.)
תַּמְשִׁילֵהוּ בְּמַעֲשֵׂי יָדֶיךָ
tamšīlēhû bĕ-maʿăsê yādeyḵā
“You made him rule over the works of Your hands”
Humanity’s dominion in creation
Ps 19:14 (19:13 Eng.)
אַל יִמְשְׁלוּ בִי
ʾal yimšĕlû bî
“let them not rule over me”
Prayer against sin’s mastery
Ps 22:29 (22:28 Eng.)
וְהוּא מֹשֵׁל בַּגּוֹיִם
wĕhûʾ mōšēl ba-gôyim
“He rules over the nations”
God’s kingship
Ps 59:14 (59:13 Eng.)
יִמְשְׁלוּ בְיַעֲקֹב
yimšĕlû bĕ-yaʿăqōb
“they rule over Jacob”
Enemies oppressing Israel
Ps 66:7
מֹשֵׁל בִּגְבוּרָתוֹ
mōšēl bi-gĕvūrātō
“He rules by His power”
Instrumental sense (“by means of”)
Ps 89:10 (8
If Adam were to have been said to rule "with" Even, 'im' would have been used:
Reference
Hebrew Phrase
Transliteration
Literal Meaning
Context / Relationship
Gen 5:22
הִתְהַלֵּךְ חֲנוֹךְ עִם־הָאֱלֹהִים
hithallēkh Ḥănōkh ʿim hāʾĕlōhîm
“Enoch walked with God”
Fellowship / partnership
Gen 6:9
אֶת־הָאֱלֹהִים הִתְהַלֶּךְ נֹחַ
(ʾet + ʿim nuance)
“Noah walked with God”
Moral companionship
Gen 13:5
וְגַם לְלוֹט הַהֹלֵךְ אֶת־אַבְרָם
(ʾet + ʿim nuance)
“Lot, who went with Abram”
Physical companionship
Gen 18:16
וַיֵּלְכוּ הָאֲנָשִׁים... וְאַבְרָהָם הֹלֵךְ עִמָּם
ʾAvrāhām hōlēkh ʿimmām
“Abraham went with them”
Cooperative action
Exod 18:23
וְנָשְׂאוּ אִתָּךְ
wĕ-nāśʾû ʾittāḵ
“They will bear [the burden] with you”
Shared labor
Lev 26:12
וְהִתְהַלַּכְתִּי בְּתוֹכְכֶם
ve-hithallaktî bĕtōḵĕḵem (related)
“I will walk among / with you”
Divine fellowship
Deut 31:6
כִּי יְהוָה אֱלֹהֶיךָ הוּא הַהֹלֵךְ עִמָּךְ
hûʾ ha-hōlēkh ʿimmāḵ
“For the Lord your God is the one who goes with you”
Support, accompaniment
Ruth 1:14
וְרוּת דָּבְקָה בָּהּ
(ʿim understood)
“Ruth clung to her [Naomi]”
Loyal companionship
1 Sam 10:26
וַיֵּלְכוּ עִם־שָׁאוּל
wayyēlĕḵû ʿim Šāʾûl
“They went with Saul”
Political allegiance