As often happens when crossing linguistic boundaries, the single English verb to serve has several translations in Hebrew, corresponding to several different meanings of the word in English.
להגיש can also mean to submit something, as in:
הִגַּשְׁתִּי מָעֳמָדוּת לַתַּפְקִיד.
I applied (submitted candidacy) for the position.
An example of לשרת:
הִיא שֵׁרְתָה שְׁנָתַיִם בְּחֵיל הַיָּם.
She served (for) two years in the navy.
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