As news of Mr. Trump’s budget begins to sink in across the country, Americans are trying to parse what the changes to the government’s spending plan might mean for them. It is only a proposal, an opening bid in what is likely to be a protracted public argument over national priorities. But it is important because it signals what the new president is thinking, his wish list for the size and shape of government.

In two days of interviews with beneficiaries of programs at risk in 11 states, many people said they did not see themselves reflected in Mr. Trump’s vision for the government. And some felt surprise at what has been left out.

Ms. Feltner said that without the heating subsidy she would probably have to move in with her daughter and two teenage grandchildren. “I’d still like to have a little dignity left, and not have to move in with someone else,” she said. “I used to be the one packing up the food in the food pantry for people,” she said. “Now I’m the one in line.”

Another proposed cut would defund the Appalachian Regional Commission, which was founded in 1965 to strengthen economic growth in a 13-state swath of the country. Of the 420 counties in the commission’s footprint, 399 voted for Mr. Trump.

“I hate to see him cut us,” said Chris Farley, 32, of Delbarton, W.Va., who was laid off from his job operating a drill at a surface coal mine in 2015 and is now in a jobs and education program partially funded by the commission.