Young females always do OK but it doesn't last because they are terribly protective of what they consider their nest and, as the cage becomes just that, one of them will fight the other for it so, if I were you, I would get them tested asap. It's not hard and it's not expensive so there is no reason not to do it and a VERY big one for getting it done.
Don't sit right next to them. Give them room. Sit in the same room but not next to them. You are, most likely, scaring them with your close presence. These are not birds that have imprinted to humans so a giant alien close to them and, most likely, looking at them is not reassuring at all and you need to make them relax so take it easy and give them some room.
Wait until you get their cage, set it up and move them in. If you are free-feeding protein food, change the diet and give them something like gloop, chop or mash in the morning and seeds or pellets for dinner. If you do that, you will be able to allow them to come out to fly before dinner and they will go back into the cage on their own when you serve it. BUT for this to work out, you need to establish the cage as theirs [and that's why they need to be in their 'final' cage and not a temporary one], get them used to a completely non-changing daily routine and your presence [this takes about 10 days to two weeks] and their diet has to be right [because if they have access to protein food during the day, they would have no incentive to go back into their cage]. All this takes a few weeks so, for now, forget about letting them out of the cage but, if you decide to do it anyway, don't do it in the bathroom which is a completely unfamiliar place for them. As to the mosquito screen... yes, lovies will chew right through it so you have to keep a VERY alert eye on them, especially since you don't know if you have a male/female pair [it's usually the ones without mates that try to 'escape' all the time - because they want to go out and look for one]. When I had the lovies flock in Pennsylvania, I had to get them redone about every two weeks or so and I remember the employee of the hardware store that did it asking me if I had a bear for a dog that could make holes so fast in them to which I replied that it was no huge dog chewing the screens, it was teeny tiny birds doing it