I've heard that some people only write わたし in kana and only use the Kanji 私 when the intended reading is わたくし. How common is this practice? Is it old-fashioned? If I used it myself as a foreigner, would it seem odd?
Almost nobody cares if you write them in hiragana or kanji.
Theoretically, the kanji 私 is only associated with わたくし, whose original meaning is "personal, private". But insisting so in this age only sounds very much like the "spelling police", because the most prevalent usage of わたくし is, after all, as first-person pronoun like its shortened form わたし.
Plus, the use of わたくし is considered as a "deluxe edition" practically confined within formal business protocol and by no means is the primary choice in everyday life these days. If your company boss commands you to use 私 only for わたくし, maybe you should follow. Otherwise, we just decide how to read 私 according to the context, so you don't have much to worry about.
Of course if you really want to clarify if it's わたし or わたくし, you could make both into kana or use furigana (私【わたし】—私【わたくし】). Anyway, you couldn't expect much people to disambiguate them by kanji 私【わたくし】 vs. kana わたし.