You're probably not meant to give mishloach manos. See this Peninei Halakha for an explanation. Purim on the 14th Adar never falls out on Shabbos nowadays, but Shushan Purim in Jerusalem on the 15th can sometimes so the same issue occurs. This article explains the differences in detail. I will summarise. The same material is covered in Yalkut Yosef 688:30.
On Sunday, they eat the festive Purim meal and send mishloach manot to one another, since mishloach manot is connected to the meal... Since it is inappropriate to make such a meal on Shabbat, we postpone it until after Shabbat. It is best to eat more meat and drink more wine than usual even on Shabbat, because some authorities maintain that the seuda is supposed to be eaten on Shabbat. If possible, it is also good to send mishloach manot privately on Shabbat, on condition that there is an eruv in one’s location
See Orach Chaim 688:6: most hold like the Shulchan Aruch that the seuda is postponed to Sunday, so the mishloach manot is as well. Taz (sk.8) and Baer Hetev (sk.7) bring Ralbach (32) who disagrees and holds both on Shabbos. The reason why he says this is because, although the Yerushalmi (Megillah 1:4) suggests that the seuda should be postponed, the mitzvah is so important and clearly on the 15th of Adar that it cannot be moved. Even then, he says "והנני מודה שטעות גדולה היא אלי לחלוק על זה מכת השכל כיון שבירושלמי נאמר", he understands that it is very controversial for him to disagree with the Yerushalmi, but he feels compelled. He also argues that his view is hinted at in the Rambam, who says nothing about moving the seuda to Sunday. He also suggests that there are other ways to interpret the Yerushalmi, although the Ran 3a is clear about it meaning that the seuda should be on Sunday.
Magen Avraham (sk.10) overrules this opinion. L'maaseh most authorities do mishloach manos only on Sunday and disagree with the Maharalbach. Mishnah Berurah (688:18) holds like this (citing Radbaz 1:508 and Korban Netanel Megillah 1:7:4), and so does the Aruch Hashulchan (688:17). See Noda Biyehuda (1:42) for an explanation of why he hold like this, as well as Tashbetz 3:298.
However, the Pri Megadim EA 688:7 brings the Pri Chadash who tries to resolve this by doing the seuda and mishloach manos on Shabbos and Sunday. There is a possibility that the Magen Avraham wasn't entirely censuring mishloach manos on Shabbos, just saying that we are not obligated. Rav Moshe Sternbuch Teshuvot VHanhagot 3:239:3 says that this is the custom in Jerusalem to be careful for both opinions and agrees with the Maharalbach's reasoning.
Interestingly, this is an example of a halacha only sourced in the Yerushalmi: Talmud Yerushalmi Halachot
