It sounds like you already know the recommended way to do cat introductions, but if you can't feed across the door, start bigger. Put the food farther and farther back from the door until you find where they're most comfortable eating, feed them there for a while. And when the get comfortable eating at that distance, slowly start bringing the food closer (make sure to take it slow and be aware this could take a while) until they're finally completely comfortable eating at the door. And to deal with the new cat's wailing, you could try sleeping in that room with her, at least until you can safely let her out. And to speed up the process, have one person in there with her while she eats and one person with the resident cat while he eats (these people should be someone each cat is comfortable with). These people can function to coax the cats to eating closer to the door than otherwise possible, just be sure not to challenge them too much when not ready (you want to challenge them a little, but don't try to get them to eat at distance they're terrified of being at or start getting aggressive and focused on other cat.
But if that doesn't work or solve your problem... Letting her explore the house might actually be beneficial, she's not just getting the lay of the land but also smelling and getting used to the scent of the new cat. You can try putting toys or bedding the other cat loves to be around and putting them in with the cat who's locked in a room (that can work with both resident or new cat). And litter boxes might be especially good for this, but if your cat's not ready for that yet make sure they still have their own personal litterbox in there with them. It's also important to know that a certain amount of aggression is normal as dominance hierarchies are worked out. This is what I call false charges and typically looks like a cat with arched back, bushed up fur, and all they do is run at your other cat but will stop short of actually attacking (expect many of these false charges in a row if this is what your new cat is doing). However, if your resident cat runs or reacts aggressively, this might trigger the new cat to actually attack as they now know they have the upper hand. The best response by your resident cat is to do absolutely nothing and let the new cat tire herself out.
But if actual attacking is happening, then it is imperative you stop it now. It matters little if they false charge or hiss/growl at each other, but if one is guarding areas, staring each other down, chasing, and god forbid actually making physical aggressive contact with each other, this can ruin the relationship.
It is possible to "introduce" cats without separating them in rooms (and what I'm currently attempting with my own cats), but the method I'm about to describe is considerably more dangerous and only something you should attempt if you trust/know your cats (which might be a problem with your new cat). First off, no aggressive attacks should be allowed and any aggression should be diffused (especially on part of the aggressor, feel free to use this method on one cat at a time). If your new cat attacks your other cat or looks like they're about to attack, stop it (and this is when it gets dangerous). For my own cats, I physically hold her back and with a stressed out cat, this could lead to you getting attacked instead. I put my hands on her chest, ready to stop her if need be and I also physically try to lead her away. But if possible a safer method might be just to distract them away from the object of their aggression with things such as toys (this might be harder to do if your other cat is growling or making aggressive noises to defend themselves). I also found speaking to my cats in a loud stern voice helps too, it functions as a command to make them stop. Now all I do is repeat the same phrases in the same tone when I see my cat about to attack, and after enough time (and if she's not TOO wound up), she'll move away on her own. If your cat is too wound up to listen to those commands, its imperative your physically stop her again as any aggressive attacks can set this method back. Now to help with your defending cat (i.e. resident cat). They are acting aggressive cause they have learned to fear your new cat and come to expect aggression (which is why its so important to retrain your cat to not be aggressive). So when your new cat is ready for it, start petting your resident cat and keeping them calm while you encourage new cat to get close. But make sure you can hold your new cat back if need be and pet her too so that she can associate pleasure with being around the resident. Feeding also can be used with this method and try feeding your cats when they can get this close to each other (but you may still have to pet and comfort resident at same time, its important to keep her calm and to keep her from making aggressive/defending vocalizations). Do this as often as possible and keep doing this to resident until they stop immediately getting defensive at just the presence of your new cat. Both steps have to be done together at the same time for best results, and make sure to always listen to your cats and know when you're pushing them past their comfort zone.
One last thing I'd like to add that might help is to get new cat tress. Not only does this increase available territory, but the cat trees would be new and so don't "belong" to anyone just yet and with any luck and come to be owned by the new cat an so they can feel like some of their territory space belongs just to them and no one else (right now, your house is probably being perceived as belonging to your resident cat and leaving the new cat to not feel like anything in the house is just theirs). Also, in case any aggression is play aggression on the part of your new cat, you can try to do heavy play sessions before introducing the two to tire your new cat out. Even if its not play aggression, tiring the new cat out before momentary introductions might be beneficial to have these meetings go well.
