My chatan and I would like to marry before Pesach. But are there any restrictions about the sheva brachot overlapping with the Chag?
It is permitted to get married before Pesach, though you should be careful of coming too close, because people are very busy with preparing for Yom Tov.
If somebody gets married before Yom Tov, the sheva berachot continue into the days of Chag, and this is no problem.
Seder Night constitutes a certain difficulty, because it is forbidden to add an extra cup of wine over and above the four cups, whereas the Ashkenaz custom is to use a separate cup for the sheva berachot. This question is discussed in the Pischei Teshuvah (Even Ha-Ezer 62:18), who quotes from the Yaavatz that the berachot should be made over the cup of birkat ha-mazon, but cites from the Rema (Darkei Moshe, Orach Chaim 473:5) that one should make all seven blessings on a separate cup. See also Iggrot Moshe, Even Ha-Ezer 1:98.