FORT WORTH — Paul D. Donahue and his wife, Angela, are among more than a million Americans who have signed up for health coverage through the federal insurance exchange. Mr. Donahue has a card in his wallet from his insurer to prove it. But when he tried to use it to get a flu shot and fill prescriptions this week, local pharmacies could not confirm his coverage, so he left without his medications.

Similar problems are occurring daily in doctors’ offices and drugstores around the country as consumers try to use insurance coverage that took effect on Jan. 1 under the Affordable Care Act.

In addition to the difficulties many face in proving they have coverage, patients are also having a hard time figuring out whether particular doctors are affiliated with their health insurance plan. Doctors themselves often do not know if they are in the network of providers for plans sold on the exchange.

But interviews with doctors, hospital executives, pharmacists and newly insured people around the country suggest that the biggest challenge so far has been verifying coverage. A surge of enrollments in late December, just before the deadline for coverage to take effect, created backlogs at many state and federal exchanges and insurance companies in processing applications. As a result, many of those who enrolled have yet to receive an insurance card, policy number or bill.