That must be why you want a bird!
Sorry, I just couldn't resist!
Welcome! You will find lots of great information and support from caring experienced individuals here!
One problem with parrots and small children is that parrots (and little kids) are unpredictable. Kids can move quickly and let out sounds that startle parrots, even a budgie can inflict a bite that would upset or hurt little fingers.
Another problem, mostly with parrots larger than a cockatiel, is many of them have distinct preferences a certain person in a household. Some species more than others but none are always accepting of everyone.
The desire for a "cuddly" parrot is understandable, especially if you watch YouTube videos of perfect parrots cuddling and asking for head scratches. Most are babies, and most baby parrots are cuddly. Most adult parrots are not. Parrots generally don't like to be petted anywhere but on their heads, necks and faces, only when they feel like it, and only a "certain way", depending on the individual bird. Babyhood lasts as short as two months in budgies to as long as a year or two in the largest parrots. When puberty strikes, all hell can break lose as they're flooded with hormones that can make them irritable, unpredictable, and sometimes dangerous.
I haven't said a word about Senegals or related birds because I've never had one or known one. Read up as much as you can in these forums and elsewhere for species specific characteristics. What I've read about Senegals is that if you acquire a young hand raised one they CAN BE sweet affectionate fun little parrots that like some "cuddling", like snuggling up to your neck while riding on your shoulder and accepting head scratches. That's the extent of cuddling you can expect. As for being one person birds, I read that some are and more aren't, but that's very unpredictable, and the person they like most may change when they hit puberty. Older birds at rescues aren't necessarily "problem birds". Their owners may have died or become unable to take care of them.
I had a hand rsised cockatiel years ago that would have been the perfect bird for your household. I'd consider one if I were you, especially while your children are so young.
You should take your time and consider your expectations carefully before adopting any parrot.