Americans have known two times in which one of the two main Parties die: the Federalists and the Whigs both in the early half of the 19th century. In both cases the Democratic Party became the 'only game in town', and elections were decided during the primaries instead of during the general election. In both cases the Democratic Party split because it was unwieldy. The Whigs formed the first time, and the Republican Party formed the second time.
Yes, the Federalists were last on the presidential ballot in 1816 and for the next two cycles only Democratic-Republicans were on the ballot. However there were no primaries for the D-Rs and nominees came from Congress. In 1824 that system failed and the electoral college acted as a primary with the runoff occurring in the House. In 1828 the D-Rs split with each separately nominating a candidate. The Whigs were not the immediate product since in 1828 and 1832 the Democrats and National Republicans were the parties. The Whigs emerged from a merger after the 1832 elections. And the most significant change was the use of nominating conventions to select presidential nominees.
The Republicans of the 1850's did not form from a split within the Democratic Party, but from a split within the Whigs over the issue of slavery. The three candidates in 1856 represented the Democrats and two former Whig factions one of which was the Republican and the other was headed by former president Fillmore, and the two former factions again had candidates in 1860 (Lincoln and Bell). So it is quite incorrect to say that the Democrats ever became the 'only game in town' at a national level after the initial split by 1828. Also it is incorrect to think modern primaries decided anything in the 1800's since they weren't in use until the 20th century.
