There are no "exact dates for the state of war" because that's an anachronistic way of looking at relations in antiquity. Instead, you have to view their relations through their surrounding context.
Moreover, all this information is easily available in standard histories of the time period. I'll cite Amélie Kuhrt's The Ancient Near East : c.3000-330 BC (1995) and Ian Shaw's Oxford History of Ancient Egypt (2000), which I have available at home. I'm sure more recent historians have offered more precise dates.
I'll begin with a bit before this timeline with the rise of the Hittites. Suppiluliumas "early in [his] reign" (Kuhrt 253) was fighting the Mittani, who at that time were allied with Egypt and whose leader, Tushratta, was Amenophis III's brother-in-law. The Mitanni were also up to this point a major player, and their alliance with the Egyptians is one of the reasons that Egypt receded from warfare in Syria (Kuhrt 194). Kuhrt also posits that the growing Hittite empire probably helped secure the alliance.
That said, after the Hittites' successes, the Egyptians tried to broker peace with them instead. During the reign of Tutankhamun (likely, could be earlier), tried to establish a peace treaty with the Hittites. What need for a peace treaty would there be if they were already at peace, right? So they must have feared the Hittites' continued dominance in the region, especially after the Hittites took over what was formerly Egypt's vassals in the Levant (Shaw 282).
Relations weren't perfect between the two, though. When Suppiluliumas received a request by Tutankhamun's widow Dakhamunzu (Ankhesenamun?) to marry one of his sons, he was very wary that it was a trap, and indeed his son was murdered when he got to Egypt, perhaps by a faction associated with Ay, who took over after Tutankhamun's death (Kuhrt 254, Shaw 283-284). You can read about this exchange on Wiki. The question remains: Why would Suppiluliumas distrust so much if he had peaceful, friendly relations with Egypt?
This takes us down to the beginning of Ay's reign in 1327 (following Shaw's chronology). After this, the "peace" was shattered. First, a plague ravaged the Hittites and killed both Suppiluliumas and his son Arnuwanda. This allowed for the Egyptians to once again campaign in the Levant and take back territories lost to the Hittites:
Trouble with Hittites over territories in northern Syria continued, and around regnal year 10 (of Horemheb's reign, so 1313 BCE) the Egyptians appear to have made an unsuccessful attempt to reconquer Qadesh and Amurru, although it is typical of the reign that our sources for this confrontation are Hittite. (Shaw 284)
So, in 1313, according to Shaw, the Egyptians and the Hittites are engaged in hostilities. The Hittites don't mention any weird happenings in Egypt at this time. Even using revised chronologies, we still have hostilities during this time period and an ascendant Egypt who is trying to conquer much of Syria.
Sety I (1294-1279) further campaigned against the Hittites:
Sety began in his regnal year I [=1294] with a relatively small-scale campaign against the Shasu in Palestine, soon followed by military expeditions further north. In a later war he moved into territory held at the time by the Hittites... The result as a war with the Hittites during which both vassal states [=Qadesh and Amurru] were lost again, followed by a period of guarded peace.
Following Sety's death, in his fourth regnal year (1275), Ramesses II, Sety's successor, and Muwatilli II fought at the battle of Qadesh. What followed was a truce, not a peace, and they wouldn't have a true peace treaty until 1259 or 1258 with the Treaty of Kadesh.