Ya'akov Avinu from the Parsha of Toldot through to Vayishlach goes on a transformational journey. He begins as a quiet shepherd, seemingly content with a simple life. We also see him in what may be seen as deceptive behaviour, tempting his brother into selling his birthright and deceiving his father into believing that he is Esav. The following sources are an exploration of the word ויגש and its use with respect mainly to Ya'akov.
The word ויגש appears 9 times in the Torah [based on a Sefaria.org search]. 8 of them in Bereishit and once in Vayikra.
(כג) וַיִּגַּ֥שׁ אַבְרָהָ֖ם וַיֹּאמַ֑ר הַאַ֣ף תִּסְפֶּ֔ה צַדִּ֖יק עִם־רָשָֽׁע׃
(23) And Abraham drew near, and said: ‘Will You indeed sweep away the righteous with the wicked?
(א) ויגש אברהם. מצינו הגשה למלחמה (ש"ב י יג) ויגש יואב וגו' , הגשה לפיוס (להלן מד יח) ויגש אליו יהודה, והגשה לתפלה (מלכים א' יח לו) ויגש אליהו הנביא, ולכל אלה נכנס אברהם, לדבר קשות, ולפיוס ולתפלה:
(כב) וַיִּגַּ֧שׁ יַעֲקֹ֛ב אֶל־יִצְחָ֥ק אָבִ֖יו וַיְמֻשֵּׁ֑הוּ וַיֹּ֗אמֶר הַקֹּל֙ ק֣וֹל יַעֲקֹ֔ב וְהַיָּדַ֖יִם יְדֵ֥י עֵשָֽׂו׃ (כג) וְלֹ֣א הִכִּיר֔וֹ כִּֽי־הָי֣וּ יָדָ֗יו כִּידֵ֛י עֵשָׂ֥ו אָחִ֖יו שְׂעִרֹ֑ת וַֽיְבָרְכֵֽהוּ׃ (כד) וַיֹּ֕אמֶר אַתָּ֥ה זֶ֖ה בְּנִ֣י עֵשָׂ֑ו וַיֹּ֖אמֶר אָֽנִי׃ (כה) וַיֹּ֗אמֶר הַגִּ֤שָׁה לִּי֙ וְאֹֽכְלָה֙ מִצֵּ֣יד בְּנִ֔י לְמַ֥עַן תְּבָֽרֶכְךָ֖ נַפְשִׁ֑י וַיַּגֶּשׁ־לוֹ֙ וַיֹּאכַ֔ל וַיָּ֧בֵא ל֦וֹ יַ֖יִן וַיֵּֽשְׁתְּ׃
(22) And Jacob went near unto Isaac his father; and he felt him, and said: ‘The voice is the voice of Jacob, but the hands are the hands of Esau.’ (23) And he discerned him not, because his hands were hairy, as his brother Esau’s hands; so he blessed him. (24) And he said: ‘Art thou my very son Esau?’ And he said: ‘I am.’ (25) And he said: ‘Bring it near to me, and I will eat of my son’s venison, that my soul may bless thee.’ And he brought it near to him, and he did eat; and he brought him wine, and he drank.
Ya'acov, usually reticent and passive understands the nature of his brother who is more pro-active. In order to convince his father that he is Esav he must act like him.
(י) וַיְהִ֡י כַּאֲשֶׁר֩ רָאָ֨ה יַעֲקֹ֜ב אֶת־רָחֵ֗ל בַּת־לָבָן֙ אֲחִ֣י אִמּ֔וֹ וְאֶת־צֹ֥אן לָבָ֖ן אֲחִ֣י אִמּ֑וֹ וַיִּגַּ֣שׁ יַעֲקֹ֗ב וַיָּ֤גֶל אֶת־הָאֶ֙בֶן֙ מֵעַל֙ פִּ֣י הַבְּאֵ֔ר וַיַּ֕שְׁקְ אֶת־צֹ֥אן לָבָ֖ן אֲחִ֥י אִמּֽוֹ׃
(10) And it came to pass, when Jacob saw Rachel the daughter of Laban his mother’s brother, and the sheep of Laban his mother’s brother, that Jacob went near, and rolled the stone from the well’s mouth, and watered the flock of Laban his mother’s brother.
This is the first time Ya'acov shows his initiative and takes control of a situation. Note how this episode can be linked with the way his mother (Lavan's sister) watered the camels of Eliezer and how Moshe watered Yitro's flocks.
(יח) וַיִּגַּ֨שׁ אֵלָ֜יו יְהוּדָ֗ה וַיֹּאמֶר֮ בִּ֣י אֲדֹנִי֒ יְדַבֶּר־נָ֨א עַבְדְּךָ֤ דָבָר֙ בְּאָזְנֵ֣י אֲדֹנִ֔י וְאַל־יִ֥חַר אַפְּךָ֖ בְּעַבְדֶּ֑ךָ כִּ֥י כָמ֖וֹךָ כְּפַרְעֹֽה׃
(18) Then Judah came near unto him, and said: ‘Oh my lord, let thy servant, I pray thee, speak a word in my lord’s ears, and let not thine anger burn against thy servant; for thou art even as Pharaoh.
An important episode with Ya'acov's children. Yehuda takes responsibility for his brother Binyamin and retroactively for the sale of Yosef.
(י) וְעֵינֵ֤י יִשְׂרָאֵל֙ כָּבְד֣וּ מִזֹּ֔קֶן לֹ֥א יוּכַ֖ל לִרְא֑וֹת וַיַּגֵּ֤שׁ אֹתָם֙ אֵלָ֔יו וַיִּשַּׁ֥ק לָהֶ֖ם וַיְחַבֵּ֥ק לָהֶֽם׃ (יא) וַיֹּ֤אמֶר יִשְׂרָאֵל֙ אֶל־יוֹסֵ֔ף רְאֹ֥ה פָנֶ֖יךָ לֹ֣א פִלָּ֑לְתִּי וְהִנֵּ֨ה הֶרְאָ֥ה אֹתִ֛י אֱלֹהִ֖ים גַּ֥ם אֶת־זַרְעֶֽךָ׃ (יב) וַיּוֹצֵ֥א יוֹסֵ֛ף אֹתָ֖ם מֵעִ֣ם בִּרְכָּ֑יו וַיִּשְׁתַּ֥חוּ לְאַפָּ֖יו אָֽרְצָה׃ (יג) וַיִּקַּ֣ח יוֹסֵף֮ אֶת־שְׁנֵיהֶם֒ אֶת־אֶפְרַ֤יִם בִּֽימִינוֹ֙ מִשְּׂמֹ֣אל יִשְׂרָאֵ֔ל וְאֶת־מְנַשֶּׁ֥ה בִשְׂמֹאל֖וֹ מִימִ֣ין יִשְׂרָאֵ֑ל וַיַּגֵּ֖שׁ אֵלָֽיו׃
(10) Now the eyes of Israel were dim for age, so that he could not see. And he brought them near unto him; and he kissed them, and embraced them. (11) And Israel said unto Joseph: ‘I had not thought to see thy face; and, lo, God hath let me see thy seed also.’ (12) And Joseph brought them out from between his knees; and he fell down on his face to the earth. (13) And Joseph took them both, Ephraim in his right hand toward Israel’s left hand, and Manasseh in his left hand toward Israel’s right hand, and brought them near unto him.
Despite Ya'acov being old and frail he is referred to as Yisrael, an indication of his inner strength which at times left him during his turbulent life, this is further emphasised by the use of the word ויגש.