First, some background information:
Hilkhoth Tefillah Uvirkath Kohanim chapter five lists eight requirements that are necessary for tefillah, but are me`aqev if not performed. Among the list [5:10] is hashwayath haqqol - "the leveling of the voice." This is explained as follows:
The leveling of the voice. How? One should not raise his voice during his prayer, and he should not pray in his heart [i.e. without words], but rather should "cut" the words [i.e. mouth them] with his lips and cause his ears to hear them in a whisper. And he should not raise his voice, unless he is ill or is not able to properly direct his heart until he causes his voice to be heard - it is permitted [in such cases] to do so. Except he should not do so in a congregation so that their minds are not disturbed by his voice.
Elsewhere, in Hilkhoth Tefillah 4:18, it says:
And thus we also do not stand to pray from the midst of joking, nor from the midst of frivolity, nor from the midst of common conversation, nor from the midst of dispute, nor from the midst of anger - but rather from the midst of words of Torah. And not from the midst of determining halakhah (although such things are words of Torah) so that his heart will not be distracted by halakhah. Rather, [we stand to pray] from words of Torah that do not require investigation, like halakhoth pesuqoth.
As for the require to properly and exactly enunciate each letter of lashon ha-qodhesh in areas of the liturgical prayers of the beth kenesseth, it is required in only three places:
- Shema - In Hilkhoth Qiryath Shema` 2:8 it says: "wa-ssarikh le-dhaqdeq be-othiyyothehah wa-im lo dhiqdeq yassa - And it is necessary to be exacting in [the pronunciation of] its letters, and if one was not exacting he [nevertheless] fulfills his obligation."
- Birkath Kohanim - In Hilkhoth Tefillah Uvirkath Kohanim 14:11 it says: "ein birkath kohanim na'amarath be-khol maqom ela bilshon haqqodhesh - In every place that the priestly blessing is said, it is not said except in the holy language [i.e. Hebrew]." And in 15:1-2 it says: "shishah devarim mon`in nesi'ath kapayim: hallashon... Hallashon kessadh? ha-'illghin she-ein mossi'in ha-othiyyoth ke-thiqnan, ke-ghon shaqqorin le-'ilfin 'aynin ule-'aynin ilfin, o le-shibbolth sibboleth wa-khayyosse ba-han, ein nos'im eth kapeyham. wa-khen kivdhei peh wa-khivdhei lashon, she-ein divreiham nikarim lakol, ein nos'im eth kapeyham. - Six things prevent the lifting of the hands: the language [used]... The language [used]. How? Those who pronounce Hebrew in an incorrect manner and do not produce the letters as they were instituted, like those who pronounce alefs like ayins and ayins likes alefs or who pronounce 'shibboleth' as 'sibboleth' and other similar things, such [kohanim] do not lift their hands. And also those who are stammerers and of unclear speech whose words are not readily recognized by all, such [kohanim] do not lift their hands."
- Hazarath HaShass by the Shaliah Ssibbur: In Hilkhoth Tefillah Uvirkath Kohanim 8:12 it says: "wa-khen ha-`illegh keghon mi shaqqore' le-ayin alef o le-alef ayin wa-khol mi she-eino yakhol le-hossi eth ha-othiyyoth ke-thiqnan ein memanin otho shaliah ssibbur - And also the one who pronounces Hebrew in an incorrect manner, like one who pronounces an ayin like an alef or an alef like an ayin, and everyone who cannot produce the letters as they were instituted, we do not appoint such a one to lead the prayers of the congregation."
In no other part of the service is this type of precision in pronunciation required, including one's whispered, personal prayer.
Now, to answer your question:
Le-`aniyuth da'ati, not only is it permissible to not emphasize the zahnim in your tefillah so that they sound in a whisper like you pronounce a samekh, but it seems that it is correct to not worry about such things at all during your tefillah. What is important is directing your heart toward the Creator.
I say this because the pronunciation of the letters is a matter of halakhah that requires a bit of investigation and the Rambam states that such matters of Torah only serve to distract us during prayers [cf. Hilkhoth Tefillah 4:18].
So, my answer and advice to you is pray the words of the tefillah, focus on their meaning and completely forget about their exact pronunciation. The halakhic imperative of the silent shemoneh `esrei is to direct your heart and not disturb the concentration of others praying around you. "Buzzing like a bee" is sure to disturb people and will not serve to make your prayer any more acceptable, but possibly - has wa-shalom - the opposite.
Hope this helps. Kol tuv.