I am a Noahide and I want to wear some kind of shawl during my personal prayers at home, understanding the importance and relevance of the tzitzit fringes to the Jewish people. I would not want to offend anyone or do the wrong thing by wearing them. I saw a nice shawl without the tzitzit fringes, so can I buy and use this one at home? (I don't want to be some kind of imposter and use it outside to possibly confuse someone if they saw the shawl.)
The religious significance in wearing a prayer shawl lies specifically in the tzitzit fringes themselves. Technically any garment of four corners can be used as a prayer shawl, so long as it has the tzitziyot on them. As for a non-Jew wearing tzitzit, there is nothing offensive or wrong with your wearing tzitzit so long as you are aware that you do so on a voluntary basis and are not obligated to do so in any way. Rambam, Hilchot Melachim uMilchamot 10:11 states that a non-Jew is permitted to perform most mitzvos for the purpose of receiving reward.
Considering the wording of the Mitzvah @ Deut. 22:12 "You shall make yourself twisted threads, on the four corners of your garment with which you cover yourself", it appears to imply that there was an already pre-existing garment, which was typical of the era, that did not have tassels or otherwise attached in any form. I think this may suggest that though it is a meritorious commandment to attach Tzitzit as a Jew, a Noachide could achieve some merit from covering himself ritualistically in a shawl, whether it be a Tallit with the Tzitzit removed (such as myself), or your grannies best knitting!