You should read my post here(
housetraining double trouble - explain this behavior!) about potty training.. this COULD be a potty training issue AND an anxiety issue
Also, I hate to say this, but have you TRIED to not use the crate? I tried a crate with my Cappy because thats what the shelter said.. then I snapped out of it and realized that I have had rescue dogs growing up and we NEVER used a crate.. crates are a thing of the new... they are more convenience for the owner than for the dog.. yes, dogs come to love their crates as their home, but sometimes its at a HUGE stress to the dog in the beginning.
Anyways.. try, instead of crating, use baby gates (or bigger if you have a large dog) and confine him to one room where he will do the least damage.. This paired with some serious potty training SHOULD help.. Yes, you have a new dog and you WILL HAVE ACCIDENTS>
if you are not prepared to deal with addidents while your dog adjusts, then you should not have a dog.. But if you are willing to put the time and effort into getting this dog situated, then he could be an AMAZING pooch! BUt it could be a big project.. Shelter dogs can have some severe separation anxiety issues.
A good way to deal with separation anxiety is to teach the dog that everytime you leave, you will come back. The anxiety is based on the fact that the dog thinks that every time you leave, the dog is alone forever..
So this is what you do to teach them that it is not the case..
with your dog out of the crate, start putting on your coat and shoes (your normal leaving ritual). If the dog starts to freak out, try and ignore him but dont leave until he calms down.. Just sit on the couch and ignore the dog until he calms down.. when he calms down, take all of the items off.. wait a few minutes and put the stuff back on and sit with it on until he calms down.. then take it off.. In no time the dog will be desensitized to this "leaving ritual" and will not react to it.
Then go a step further.. Put all of the stuff on and walk out the door.. he will probably freak like usual. This time, only stay out for 20 seconds and then come back inside with treats and happiness BUT when you come back in, ONLY give the dog the treat when he settles down and is quiet.. With the treat in hand, tell him to SIT and QUIET.. even if he sits and is quiet for a split second give him the treat RIGHT AWAY! Dogs associate the last behavior they did with the treat.. Also, i forgot to mention, use an AWESOSME treat.. something like cheese, boiled chicken, or bacon pieces (small bits).
Keep on doing this in and out of the house thing.. Eventually the dog will sit quietly at the door waiting his 20 seconds for his treat because he will associate you leaving and him being quiet with a treat..
THEN.. do the same thing but make the time a little longer and longer.. take a walk around the block.. 5 minutes... 10 minutes.. etc.. Do NOT move on to the next step until the dog has completely grasped the first step.
Once you hit the 20 minute mark, any amount of time should be fine!!!
You will have to get in and out of your house a million times, and you will feel ridiculous, but this WILL HELP!
Watch Victoria Stillwells "Its Me or the Dog" on animal planet and you can see this method in action.. She's a great dog trainer when it comes to this type of stuff