ALTHOUGH the American economy has apparently bottomed out, many of the nation's cities and towns are sinking into financial quicksand.The National League of Cities (NLC), in a new study, reports that two-thirds of all cities in the United States face budget deficits, some of them extremely serious. This squeeze on city finances affects millions of people. Across the country, schoolteachers are being fired, police ranks are being thinned, roads and bridges are going unrepaired, library hours are being reduced - and taxes are going up. Rising unemployment makes the problem even worse as cities divert scarce funds into emergency assistance for families. Problems are particularly acute in smaller cities and towns, where nearly 30 percent of all communities are grappling with significant budget deficits of 5 percent or more. "It's misleading to think the financial problems that make headlines in New York or Philadelphia are an affliction unique to the big cities," says Paul Thornton, chairman of the NLC small cities council. Mr. Thornton is a councilman in Vienna, Va. Carol Day, a councilor in Gorham, Maine, speaks bluntly about the severe challenges now confronting her community of 11,000 people. Gorham is a fast-growing college town, home to the University of Southern Maine, 18 miles west of Portland. The community is being slammed by federal and state cutbacks. This year, for example, Gorham's general fund budget will drop $1 million to $4.9 million because of reduced assistance from state and federal governments. At the same time, local property taxes are being raised. "Property owners will have to pay 5 percent more, but they are getting 25 percent less service," says Ms. Day. Budget cuts will mean shorter hours for Gorham police, delayed maintenance for roads, and defrayed capital spending. With general funds in short supply, Gorham also has problems with its school budget. Day says school spending will be slashed by $733,000 to $11.2 million. Five teachers were laid off, the number of teacher assistants will be reduced, some classes will be eliminated, and sports like wrestling will be dropped. But Gorham, like many communities, finds it easiest to save money by defraying maintenance of roads, bridges, and other big-ticket items. For the past two years, for example, Gorham has put off all road repaving, which costs $40,000 a mile. The city has 127 miles of roads, and "conservatively, 120 miles needs to be repaved," Day says. This year's budget includes only $30,000 for repaving, which means a patches of road will be done, but no major projects. "It's bubble-gum and Band-Aid," she says. "And it's going to haunt us.... We are going to go so far that we will have to rebuild the roads for twice to three times as much, for you lose your road if you wait too long." Nationwide, cities are being pinched two ways with revenues dropping just as expenditures rise. On the expenditure side, most cities (53 percent) blame rising costs of personnel for their problems, especially such factors as health care. Also hurting are the growing costs of disposing of solid municipal waste, greater infrastructure needs, and the expense of federal and state government mandates for everything from programs for the disabled to clean drinking water. Revenues are falling quickly. The recession has sent property values plunging, which hurts tax revenues. Political scientist Michael Pagano of Miami University, who wrote the NLC report, says that officials in some cities are in an untenable position. Although their needs are growing, they cannot raise their property taxes because of legal restrictions. The long-term results could be negative, Professor Pagano says. In 1991, for example, an estimated 43.2 percent of all cities will reduce capital outlays, even though the nation's need for higher infrastructure spending is widely understood. Day suggests that local governments are doing just about all they can on the revenue side. Property taxes can be raised only so far, she notes. In Gorham, a house assessed for $100,000 is taxed $2,200 per year. Already, those taxes "are pitting the young people with children against the older citizens on a fixed income," Day says. Some older citizens are "frightened and angry," as property taxes have risen 75 to 80 percent in the past nine years. The answer? Day says more revenue must flow from the state and federal governments, which control most tax sources. At the same time, those governments must stop putting so many new requirements, such as tougher drinking water standards, on local government unless they are going to pay for them. "All these ideas [like clean water] have merit, but we cannot do it with the existing funds that we have," Day says.