New Hampshire Democratic Party leaders went eyeball-to-eyeball with party officials here yesterday in an effort to make certain that New Hampshire continues to be the first state in the nation to hold its presidential primary. Nobody blinked.

Democratic National Committee Chairman Charles T. Manatt warned after the meeting with New Hampshire Democratic Chairman Richard E. Boyer that if New Hampshire holds its 1984 primary before its assigned March 6 date, the vote will be a non-binding "beauty contest," and some other way will be devised to select the state's national convention delegates.

A few hours after the Manatt-Boyer meeting, the Democratic leader of the New Hampshire House, Chris Spirou, introduced a bill in Concord that would mandate scheduling New Hampshire's primary a week before any other state's primary, non-binding "beauty contest" or caucus.

The bill would place New Hampshire's primary a week ahead of Vermont's. Vermont has angered its New England neighbor by planning to hold a presidential "beauty contest" on March 6. Spirou's bill also would put New Hampshire ahead of Iowa, which has captured a spotlight of its own as the first caucus state. Iowa's caucus is set for Feb. 27 next year.

Under party rules adopted last year to prevent this sort of one-upmanship, all 1984 primaries and caucuses are to be held within a 13-week period starting March 13. Bowing to tradition, party leaders exempted Iowa and New Hampshire from these rules, which also do not apply to the Vermont primary because delegates to the convention will not be selected at that time.

But the Vermont primary could preempt national media coverage, and that has New Hampshire boiling.

"The point is, we were supposed to have an exemption," Spirou said. "What the hell kind of exemption is it? We're being sentenced to oblivion."