Democrats and Republicans compete with each other over the votes, money, and connections of special interest groups. In order to form an organization that is capable of meeting the demands of these special interest groups, the two political parties have to develop a platform that meets an extremely tailored and abstracted range of demands and expectations which REQUIRE that individual members defer their regional goals on behalf of this platform.
If R. Hank from Minnesota wants to get anything done on behalf of behalf of his constituents, then he is REQUIRED to agree to the entire platform of the broader Republican Party, regardless of whether the people Minnesota only care about that platform in bits and pieces. If Hank resists, then he is punished by being excluded from committees that draft legislation he needs to have input on in order to represent the people of Minnesota.
As soon as the Republican Party is gone, then any pressure the Democrats feel to function as a cohesive entity will be gone, as the Democrats only function as a cohesive entity in order to remain relevant against the Republican Party.
The Democrats will probably continue existing as a weak nebulous of Congressional politics, but Independents, third-parties, and surviving politicians under the Republican label would affect their internal politics so heavily that there would be very little resemblence to the Democratic Party as it exists now.