So after watching my mom feed Alex enough times, I finally felt comfortable in trying to do it myself. I asked my mom to show me how to do it, and after walking me through it, I tried it myself.
I was afraid to go too fast, because I thought Alex might choke, so I ended up going a lot slower than my mom does it. I was successful though, and managed to feed him with the syringe.
Alex is one heck of a messy eater though. No matter how careful you might be in making sure the food gets in his mouth, some ends up dripping down the front of him. So we always have to wipe his beak and chest after we finish, and for whatever reason, he seems to think of that as the worst thing in the world (we don't hurt him doing it. He's just a bit of a crybaby).
So now I'm taking over feeding time. Funny enough, whether my mom feeds him or I do, he'll cry even when he expresses that he's full. Probably because I usually put him in the cage afterwards, and he just wants to stay out... All the time. Thing is, when he's out, Lucy has to be out too, or else Lucy will start screaming, and sometimes I just don't have enough energy to keep up with two birds (especially not in the morning). So I usually have to take a few minutes to calm little crybaby Alex down again, before putting him back.
When I put him back after feeding time, I put him right in front of the food bowl, because he likes to eat after he's been syringe fed (no idea why). It's kinda funny, because he'll literally bury his face in the pellets before getting one out. He's quite capable of eating them, as well as seeds and other solid foods.
Weaning is taking longer than originally expected, but that's ok. We finally got him down to two feedings per day. Ironically, the breeder told us to originally syringe feed him twice a day, but I think the stress of moving from one place to another set him back. We ended up feeding him four times a day at first. We're finally back down to two, and I'm hoping to get him weaned pretty soon. I don't want to force wean though, because that'll cause some behavior problems in the long run.
But little Alex is doing fine. I just have a feeling he might be crying more when he sees me enter the room. That'll be fun.

I'm looking forward to when he finally is weaned.