I understand but it's actually bibically sunset that a new day begins. But be that as it may, biblically Passover was instituted as a separate feast from the days of unleavened bread. The clearest evidence is here:
Lev 23:5 In the fourteenth [day] of the first month at even [is] the LORD'S passover.
Lev 23:6 And on the fifteenth day of the same month [is] the feast of unleavened bread unto the LORD: seven days ye must eat unleavened bread.
It's right there. Passover starts the evening (sunset) of Nissan 14, the beginning of the day. The Days of Unleavened Bread start at sunset at the beginning of Nissan 15. It runs for 7 days.
That means that in order to eat the Passover Lamb in the evening of the 14th, which begins at 6pm, which is the beginning of the Jewish Day, 6pm, the Lamb had to be killed on the afternoon of the 13th, in order for the meal to be prepared and eaten on the evening of the 14th,
The lamb was to be killed at the start of Passover, on the 14th of Nissan, at sunset. The 14th day started at sunset and that's when they were to kill the lamb. They had all night to cook it and eat it. There's no scriptural support that shows they killed it on the 13th.
Exo 12:6 And ye shall keep it up until the fourteenth day of the same month: and the whole assembly of the congregation of Israel shall kill it in the evening.
They were to keep the lamb until the 14th, at sunset. Then they were to kill it that evening, the evening of the 14th, after sunset.
Exo 12:18 KJV) In the first month, on the fourteenth day of the month at even, ye shall eat unleavened bread, until the one and twentieth day of the month at even.
The 14th to the 21st is one week, the Feast of Unleavened Bread is to be 7 days, that is clear from this verse what dates define that week.
Okay, so we have one verse that clearly says Passover is one feast and the days of ulb is a separate 7 day, right?
Yet as you pointed out in this verse:
Exo 12:18 KJV) In the first month, on the fourteenth day of the month at even, ye shall eat unleavened bread, until the one and twentieth day of the month at even.
It seems that the whole thing is only 7 days. That's a contradiction.
Now for this you're going to have to do some digging because it's not obvious. But Exodus 12:6 and Exodus 12:18 actually use two different hebrew phrases that are translated "even". And Strongs concordance lists them as the same word, BUT they are not. This can be proven by looking at the scroll at blue letter bible. If you carefully look at the hebrew wording and compare verse 6 to 12, you can see it's not the same hebrew word used for "even". I'm now going to quote from a doctrinal paper our church did on the OT passover.
The term “at evening” is used twice in Exodus 12:18. What is the meaning of this phrase? The Hebrew term here is quite different from the one found in Exodus 12:6. Why was Moses inspired to use a different term? The obvious reason is that he had a different meaning in mind. The phrase “at even” in the AV comes from the Hebrew ba- erev. It generally means “sunset,” although it can refer to the time of darkness—the evening (erev). By comparing Exodus 12:18 with Leviticus 23:32 we come to the conclusion that satisfies all the scriptures. There are only seven days of Unleavened Bread and the first day is the 15th (Leviticus 23:6). The term “at even” when interpreted to be sunset can be either sunset at the beginning of a day or sunset at the end of the day. We have a similar situation with our term “midnight.” Each year on December 31 at the stroke of midnight, a new year begins AND the old year ends. Exodus 12:18 is telling us that when the sun sets at the end of the 14th, the first day of Unleavened Bread begins. The last of the seven days will end when the sun sets on the 21st day of the month. This gives us seven days of Unleavened Bread (15, 16, 17, 18, 19, 20, 21), just as we have one day of Atonement (9th day at even [sunset] until the 10th day at even [sunset]).
In other words, exodus 12:18 is counting from sunset at the BEGINNING of the 14th, to sunset at the END of the 21st...or 8 periods of sunset to sunset.
Okay, well what about not eating any leavened products on passover if it's not part of the feast of unleavened bread? The answer is that the passover meal was a sacrifice to God and anything leavened is not acceptable as a sacrifice:
Lev 2:11 Any food offering which you shall bring to Jehovah shall not be made with leaven. For you shall burn no leaven, nor any honey, in any offering of Jehovah made by fire.
This is also borne out by Lev. 6:17 and Lev. 10:12.
Man this has turned out to be one long post. Lest you think that I was smart enough to figure out all this on my own, I wasn't. I've studied that doctrinal paper our church did and it makes sense, although there's always a chance of mistakes being made in interpretation.
Doctrinal papers on this subject are available on this page.
"Passover of Exodus 12 [updated]" contains more study than you'll ever want on OT passover, while "Passover of Exodus 12 (Appendix)" contains more indepth on the meaning of the Hebrew "even" as pertains to Exodus 12:6 and Exodus 12:18.
Good luck!