USA customs and immigration

While agreeing with @Dennis regarding the alcohol (and the layover), you might need additional context for the whiskey (although it would be better as a separate question).

Entering the US, you are limited to 1L of alcoholic beverage. As long as your purchase is in a duty-free sealed bag, you can take it on your flight to the US. After going through customs in Dallas, you will have to put it in your checked luggage for your connecting flight to New Orleans.

Should you make the purchase in Manchester, its security rules note that:

Passengers will be able to buy liquids including drinks and perfumes either in an EU airport shop after security control, or on board an aircraft operated by an EU airline. If they are sold in a special sealed bag, passengers must not open them before they are screened otherwise the contents may be confiscated at the security checkpoint.

And the Heathrow FAQs explain:

New liquids rules
What are the changes to the liquids rules?
Duty free liquids, aerosols and gels purchased from any airport or airline may now be carried as hand baggage. However, these must be sealed at the time of purchase with the receipt inside a security tamper-evident bag. Following the initial x-ray scan, security officers may also need to carry out additional checks. Exceptions to the 100ml restrictions for liquid medicines and special dietary products, such as baby food, remain in place. However, following x-ray screening, security officers may also need to carry out additional checks on these items.Passengers should continue to present liquids in containers of 100ml or less in a single, transparent, resealable 1 litre plastic bag.For further information visit our Hand baggage page.

Will I be able to take my items through after any additional checks?
Yes, however where officers are unable to resolve a screening activation, exceptionally and solely for security reasons, liquids may be refused.

I purchased liquid items outside of the airport. Can I now take these through?
The 100ml restrictions for liquids remain in place with exceptions for liquid medicines and special dietary products, such as baby food.Liquids, aerosols and gels in containers over 100ml are not allowed in hand baggage unless you are carrying duty free sealed in a tamper-evident security bag, liquid medicines or special dietary products, such as baby food. Liquids not allowed in hand baggage should be placed in your hold luggage.For further information visit our Hand baggage page.

Will it take longer to go through security?
If you are carrying liquids, these may be subject to additional checks. We always recommend that you arrive at the airport as early as possible and ensure you leave sufficient time to proceed through security.

The procedure for international arrivals into the US is the same for everyone who clears immigration and customs in the US (as opposed to preclearance outside the US):

First is immigration. You'll arrive in the international arrivals area at the far end of Terminal D. Here, you queue up and go to a CBP officer to present your passport, have your fingerprints taken, etc. If you're traveling on the Visa Waiver Program and have done so at least once since 2008, you can instead use the Automated Passport Control kiosks.

Next is baggage claim. Even though your baggage tags may indicate they are checked all the way through to MSY, your bags will show up here. You must claim your bags and carry them personally through customs.

At customs, you'll make any necessary declarations, and possibly have your baggage inspected.

Immediately past customs is a baggage drop for connecting passengers. Drop your checked bags here.

After the baggage drop, you'll go through security, and then be in the Terminal D departures area. From here, you can take the airside Skylink train to your departure terminal. For a DFW-MSY flight operated by American, this could be any of terminals A, B or C, (as they are 80% of the airport's flights) so check the boards or ask airline staff.

On arrival at MSY there will be no formalities; you'll simply be pointed toward baggage claim, ground transportation and the airport exit.

To answer the last bit of your question, if you are carrying duty-free liquor from the UK in your hand luggage you'll want to move it to your checked luggage while you have the opportunity in customs. You'll be passing through security again once you recheck your bags and they'll take the bottle if you still have it.

I'd worry a bit that a1.5 hour layover when arriving at DFW on an international flight is a little tight, but if the airline ticketed that I'll assume they knew what they were doing.

You must clear US Customs and Immigration at your first Port of Entry. That would be DFW (Dallas).

You will:

1. Clear Immigration
2. Collect your checked baggage
3. Clear Customs. If you have nothing to Declare or are under the limits presented on the landing card, this is usually a formality. Don't lie.
4. Re-check your bags with American (presumably). The re-check counter us just after you exit Customs. You will have to pack the Whisky in your checked luggage.
5. Pass through TSA.
6. Board the flight to MSY.