Oh good Lordy, the spam is in all the forums.

Julia, that doggie may have been trained to go on a pee pad in the bathroom at the house of its former owner. You may want to try walking your new doggie by itself for awhile as the other dog may be a distraction. Aside from watching to see if the dog has specific times he does his potty.........my dog does............you'll have to take him out frequently and do the reward thing when he accidently does what you want him to. We have so many posts about this I don't know what to offer somebody hasn't offered already.

Thanks Calliope. I was wondering what that one post was about! I will do a search too.

Dachshunds can be more difficult to house train and require a little more patience than some other breeds, so this may be why the PO's left him in the park. Beagles fall into this same category so if you've been able to house train yours I feel sure you can house train the dachshund.

It's frustrating to bring a new pet into the house that isn't as well behaved (in house training) as your other pets, so I fully understand how you're feeling!
My advise is to pretend this is a young puppy and start house training from scratch. I would start by taking him out, without the beagle, in the yard and stay with him until he goes. Keep him moving, don't play with him and don't talk until you see him go. The instance you see him either pee or poop, give a command you want to use from now on, then give him lots of praise. I always say 'go tinkle' and 'do your business' to my dog and he will now go on command......or at least try! :~)

There is much info online regarding house training a dog so I'd advise you to read some of the articles.
I've included one that covers basics, but it may be of help.

Here is a link that might be useful: housebreaking