The thing is, there are "two" formal, "new years" in the Hebrew (Jewish Calendar). The first is defined by Exodus 12:2 which says the month of the Exodus (today the month is called "Nisan"), is the "1st month of the year". So it is okay to say "Happy New Year" on 1 Nisan, but it must be understood that it is the "New Feast Year", not the "new year" when the year is incremented in the Hebrew Calendar.
The 2nd "new year" is the 1st of Tishri. That date is Yom Teruah, and it is the 1st day of the new year, when the year number is incremented. The reason for this is primarily from Exodus 34:22, which says to "observe the festival of ingathering at the turn of the year".
The Festival of Ingathering is also known as Sukkot, and it is in the month of Tishri. So, the 1st of Tishri is the "proper" time to say "Happy New Year", only because that is when the calendar year number increments. Few people say "Happy New Year" on 1 Nisan because the year does not increment on that date.
This year, 2015 by the Gregorian Calendar, it was only coincidental that the Spring Equinox came as the 1st of Nisan began. Since the month begins in the Hebrew Calendar by the day the New Moon happens, and the Spring Equinox is determined by the position of the sun, the 1st day of Nisan will typically NOT fall on the Spring Equinox.
This year, both the traditional Jewish calendar, and the actual New Moon, said sunset, March 20th began the 1st of Nisan. The fact that the Spring Equinox was also ON the 20th of March, is not important. The important thing is that Passover is on or AFTER the Spring Equinox. And Passover, you are correct, is the afternoon of the 14th of Nisan, but that is ALSO the afternoon of April 3rd! It's the SAME day, just different calendars. So you are not correct to say April 3rd has been "thrust onto us"; it's simply the proper day but by a different calendar.
The bottom line is: This year April 3rd = 14 Nisan. (All you have to do is remember that the Hebrew Calendar date simply begins at the previous sunset by the Gregorian Calendar date because the Hebrew day begins at sunset.)
So you should have all the leaven removed from your home by early afternoon of April 3rd at the latest. The Passover meal, late that afternoon, is supposed to include matzah (unleavened bread), Exodus 12:18.