(1) How many litter boxes do you have?  Many cats will not use the same box
for pee and poop (and it may change from time to time -- where sometimes
they accept one box and sometimes not).  If you only have one litter box,
get a second box and use good-quality scoopable litter.
(2) Try feliway.  It's not a cure-all, but it does help with behavioral
problems.
(3) Can you cat see any other cats outdoors?  This can cause stress.
(4) Try to reduce the contact your children have with the cat except for
times when you can supervise them (until they are a little older and know
how to play gently with a cat.)
(5) Unfortunately, the damage has already been done -- but approximately
one-third of declawed cats develop behavioral problems such as inappropriate
peeing, biting, and scratching.  Again, try feliway and perhaps behavior
modification suggestions on the Internet.

MaryL

If it were medical, I'd gladly get it

Once again, a follow-up to my original message (now that I have given your
problem some additional thought)...

First, did your vet actually check your cat's urine, or was this guesswork?
If there is an infection and the cat was not tested, it is possible the
wrong medication was prescribed.  You might also consider the possibility
interstitial cystitis, which is a chronic inflammation of the bladder that
will mimic urinary tract infection.  This condition is exacerbated by stress
but can be controlled by putting the cat on cosequin, a canned-only diet,
and reducing stress in the home.

Are you sure the cat was spayed?  Was this done before you adopted her
(which might mean that you were mistakenly told that the cat was spayed)?
Even if you had it done yourself, it is possible some tissue was left.
Inappropriate urination, meowing at night, and her desire to get outside
could all be signs of a cat in heat.  I would certainly ask my vet to look
into this possibility.

If it turns out that the cat is spayed and there is no problem with tissue
that may have been missed, it could be that this is declaw-related behavior
and your cat could benefit from some medication.  My sister had a cat with
similar problems.  In fact, it was much worse -- she dealt with a male
(neutered and declawed) cat that sprayed constantly for more than two years.
After the vet placed him on Prozac, all of the problems literally
disappeared.

Whatever you do, please try to find out what is wrong and deal with it.  If
you return the cat to the shelter for peeing-related problems, it is likely
that they will consider the cat unadoptable and kill her.

I hope this helps.
MaryL

I got Grizzy from SSPCA 18 months ago and she had similar problems. i did
get Feliway and although it took time it did help the wet carpet problem.
She also howled all night and it did stop in itme. Th ehowling started again
a few weeks ago after my husband left and i am glad to say that with more
feliway and a lot of reassurance it is finally stoping. I think there are
time thaty Grizzy just gets too stressed and her way of communicating is to
howl.

Hope the problem resolves some way

Cati

First off, you did not mention if your vet did a urinalysis or bloodwork on
your cat.  I would like to assume at least a urinalysis was done.  If it's
not a UTI it could be diabetes (though if vet did urinalysis, should have
caught it), liver problems or kidney problems.  I suggest consulting your
vet again.

Declawed cats should NOT be left outside, they are at a distinct
disadvantage if they run up against other cats or predators.