Basset Hounds were developed to dig. Because of this genetic loading the behaviour is a strong and persistent desire. Whenever you see them digging, you can tell them "NO" or you can spray them with a water bottle, or throw a can full of coins near them (not at them). When they stop digging, reward them with praise and a toy or something appropriate to distract them. Busy dogs are less likely to dig, so taking them out to a park, or for a walk may help to curb this behaviour.
If possible, you could provide them with an appropriate designated area in your yard where they can dig. Being Basset Hounds, they would probably appreciate the opportunity. Any areas that you wish to keep Hound-free can be decorated with rocks and chicken wire to dissuade digging activities.
With your leash training attempts, make sure that you are rewarding your puppies for positive behaviour. When they walk nicely, praise them and offer them a small treat. It might be best if another person can help as you have two puppies. This requires patience and time, but keep working on it.
Remember when using treats to train, cut back on their food at meal time. Basset Hounds are particularly prone to being overweight.
* Just a side note. It is important when raising litter mates to ensure that they both bond to their human family members. Litter mates have the tendency to bond to each other more closely and therefore often continue to demonstrate more of their doggy behaviour.
Hope this helps.
Sincerely Zena & Zippy