As not saying God or writing God are newly formed ideas that many consider to be from gentile sources, it should be considered a personal stringincy, and therefore the person who was taken on this additional stringincy should do what they feel is best. If they want to know what the majority of halakhists say regarding this manner, see the following:
We follow here the ruling of the Siftei Kohen, the great 17th-century commentator to the Shulchan Arukh: "The Name of God in Hebrew is properly considered a holy name. The Name of God written in any other language, however, is not a 'holy name' at all. You will understand this when you consider that it is permissible to erase a Name written in some other language, such as the word Gott in Yiddish or German" (to Yoreh De'ah 179, no. 11). For this reason, Rabbi Yosef Dov Soloveitchik used to say that "those who write the English word God in the form G-d do so out of 'total ignorance' (am-ha'aratzut gemurah)... since the English word God is not one of the formal Divine Names but merely a literary device that refers to the Holy One, Blessed be He"; R. Zvi Schachter, Nefesh Harav (Jerusalem: Reshit Yerushalayim, 1994), 161. True, there are authorities who dispute the Siftei Kohen (see R. Avraham Danzig, Chokhmat Adam 89:9) and who support the custom of writing the Divine Name as G-d (see R. Chaim Ozer Grodzinsky, 20th-century Lithuania, Resp. Achi`ezer 3:32). We, however, following Maimonides and the other scholars we have mentioned, regard that custom as an unnecessary stringency.
Although the information is from Orthodox sources, the website is reform which apparently is problematic. Here are some other sources:
A. Opinion of the Mishnah Berurah. The Mishnah Berurah (85:10) rules
that there are no limitations placed on the writing of God’s name in
languages other than Hebrew. Indeed, this lenient view is strongly
supported by the p’sak of the Shach (Yoreh Deah 179:11) and Achiezer
(3:32). Furthermore, a paper that says “God” may be thrown away in the
normal fashion. However, the Mishnah Berurah writes that the word
“God” should not be uttered in the bathroom or other areas that are
unclean and mentioning the word “God” may be a violation of saying
God’s name in vain.
The Mishnah brerurah starts the trend of being more stringent on the name being spoken of in vain. However, it is mostly limited to bathrooms, and it "may" be saying God's name in vain.
B. Opinion of Rav Chaim Ozer Grozinski and Rav Akiva Eiger. Not only
may the printed word “God” be written and thrown away normally, but the
recital of the word “God” does not pose any halachic problem. After all,
the status of the written word should be no different than the status of the
However, if you really want to be stringent in the matter of not saying God's name, then maybe you should follow the talmudic dictum?
Shalom. The gemara (Shabbat 10b) rules that one may not greet his friend
with the word “Shalom” in the bathhouse because “Shalom” is one of the
names of God. Tosafot (Sota 10a) rules that for this reason one may not erase
the word “Shalom”. Although God is also called a “Chanun V’rachum” all
agree that these words may be erased because they are descriptions of God
rather than the formal name of God. Rosh (Teshuvot Harash 3:15) disagrees
with Tosafot and maintains that one may erase the name “Shalom” just as one
may erase “Chanun V’rachum”.
A. Writing Shalom.
1. The stringent approach. Rama (Yoreh Deah 276) cites those who
are careful not to write the entire word “Shalom”, and instead
merely omit the letter “mem” when writing “Shalom” in Hebrew.
For sources from the Talmud i couldn't find them directly translated into English so here is a paraphrased from the Jewish Encyclopedia
Talmud (Shev. 35a) lay it down that it is forbidden to erase the name of God from a written document, and since any paper upon which that name appears might be discarded and thus "erased," it is forbidden to write the name explicitly. The Talmud gives an interesting historical note with regard to one aspect of this. Among the decrees of the Syrians during the persecutions of *Antiochus Epiphanes was one forbidding the mention of the name of God. When the *Hasmoneans gained the victory they not only naturally repealed the decree, but demonstratively ordained that the divine name be entered even in monetary bonds, the opening formula being "In such and such a year of Johanan, high priest to the Most High God." The rabbis, however, forbade this practice since "tomorrow a man will pay his debt and the bond (with the name of God) will be discarded on a dunghill" the day of the prohibition was actually made an annual festival (RH 18b). It is, however, specifically stated that this prohibition refers only to seven biblical names of God. They are ʾEl, ʾElohim (also with suffixes), "I am that I am" (Ex. 3:14), ʾAdonai, the Tetragrammaton, Shaddai, and Ẓeva'ot (R. Yose disagrees with this last, Shev. 35a–b). The passage states explicitly that all other names and descriptions of God by attributes may be written freely. Despite this, it became the accepted custom among Orthodox Jews to use variations of most of those names in speech, particularly ʾElokim for ʾElohim, and Ha-Shem ("the Name" and, for reasons of assonance, ʾAdoshem) for Adonai. The adoption of Ha-Shem is probably due to a misunderstanding of a passage in the liturgy of the Day of Atonement, the Avodah. It includes the formula of the confession of the high priest on that day. Since on that occasion he uttered the Ineffable Name, the text has "Oh, Ha-Shem, I have sinned," etc. The meaning is probably "O [here he mentioned the Ineffable Name] I have sinned," and from this developed the custom of using Ha-Shem for ʾAdonai, which is in itself a substitute for the Tetragrammaton (see also Allon, Mehkarim, 1 (1957), 194ff.; S. Lieberman, Tosefta ki-Feshutah (Mo ʾed), 4 (1962), 755). *Shabbetai b. Meir ha-Kohen (first half 17th century) states emphatically that the prohibition of erasure of the divine name applies only to the names in Hebrew but not the vernacular (Siftei Kohen to Sh. Ar., YD 179:8; cf. Pitḥei Teshuvah to YD 276:9), and this is repeated as late as the 19th century by R. Akiva Eger (novellae, ad loc.). Jehiel Michael Epstein, however, in his Arukh ha-Shulhan (ḤM 27:3) inveighs vehemently against the practice of writing the Divine Name even in vernacular in correspondence, calling it an "exceedingly grave offense." As a result the custom has become widespread among extremely particular Jews not to write the word God or any other name of God, even in the vernacular, in full.