Just months into the tenure of a new party leader, Ken Martin, the Democratic National Committee’s financial situation has grown so bleak that top officials have discussed whether they might need to borrow money this year to keep paying the bills.

Fund-raising from major donors — some of whom Mr. Martin has still not spoken with — has slowed sharply. At the same time, he has expanded the party’s financial commitments to every state, and even to far-flung territories like Guam.