Firstly, good on you for getting a little rescue doggie!!
Oh dear, yes it sounds like we have a bit of problem at the moment. I would certainly be taking him to your local vet clinic for a physical exam and a urine test as soon as you can. There are a lot of things that can cause dogs (and cats) to inappropriately urinate within the home. It is difficult, of course, to know whether this is a new behaviour for him or not, as he has only just been introduced into your home.
Firstly, we need to rule out disease processes such as urinary tract infections, bladder stones, prostatic diseases and other metabolic diseases. These conditions can cause dogs to need to urinate frequently or large volumes, which can often lead to inappropriate urination in the house.
Once we are happy that there is no medical cause for this, then making a plan with your vet to establish a training regime that will work for you and your little mate would be useful. If he is also suffering significantly from anxiety, treating this as well may assist with his problem.
I hope you can find a solution and we can get him back on track again in his wonderful new home! I am hopeful that if you can find the underlying cause that this can be corrected and you can all live a happy life together. Let us know how you go with your cute little Jack Russell.