I doubt this scenario would appeal at all to the American public. America began regretting its expansion outside the Western Hemisphere fairly quickly after the Spanish-American War.
The Japanese are very unlikely to give up the Pacific territories it occupies, and the other Pacific territories will be given to Australia. Out of all those possessions, probably only Western Samoa would be given because the US already occupies some of Somoa already.
So that leaves the German colonies in Africa. If there is any place the US does not want to be involved it, it would be Africa. The US was reluctant to be involved with Liberia. Plus some of those African territories will probably be kept by the British (Namibia for South Africa, and Tanganyika to provide a land link from Cairo to Capetown).
So this isn't very realistic.
But simply to play the game, what would a US administration actually be like (at least for Togo and Cameroon)? First, unlike the British and French, the US would probably intend to administer things so that the colony becomes independent. It would probably be a cross between how the Philippines developed and how the Marines ran Haiti.
On the plus side, the US would probably invest in infrastructure in the colonies using whatever revenues the colonies generated. I don't think much would be spent from the general revenue of the US though. Commercial agriculture would be developed, probably with strong investment by US businesses. Some kind of basic education would be set up, but probably concentrating on vocational training. Some kind of liberal arts and professional training would need to be done though since that would be needed for an elite to take over.
On the negative side, the US administrators would be very racist, and would probably use labor conscription for much of the infrastructure building. The African mandates would lack most of the educated elite and instititutions that even Haiti had.
African-Americans would likely be very interested in the development of these African mandates. This would create the conditions for long term association between the US and the Mandates. Since much of the slave trade came from West Africa, such ties would likely be emphasized. This wouldn't have much impact for the first few decades, and might lead to some interesting results after the 1960s.
There would probably be legislation passed quite early guaranteeing independence of the mandates - the US would want to get rid of them as soon as possible, which would still be understood to mean several decades.
Assuming WWII still happens, the mandates would see massive investment to both provide raw materials for the war effort, and for port and airfield infrastructure to support the war.
Decolonization would proceed better for these Mandates than for the European powers simply because the US would have no interest in remaining, and this would be very clear to the indigenous peoples. Formally the mandates would transition to a UN Trust Territory. Internally, there would probably be negotiation between the new indigenous elites and the US as to how the transfer would occur. The most likely scenario would be internal matters totally turned over over to the natives with some US supervisory role in the 1950s, and control over external/foreign policy and military matters done in the 1960s. Full independence would be made before 1970.
Without a guerilla war, slightly delayed independence from OTL, and earlier efforts made to make the mandates self-governing, they'd probably be one of the better governed African countries, but that is not saying much. Both Liberia and Philippines went downhill under strongmen, and it's entirely possible that this can happen as well. But it's possible that those countries could avoid those problems, and simply by avoiding the worst would position themselves as some of the leading countries of Africa by our time.
However, if racist attitudes of the initial administrators really poison American relations with the mandates' population, then the handover would be much more difficult. But with no mulatto population to favor, I think American racism would be contained in ways to limit the damage it could do internally by dividing the native population against itself.