Greetings, welcome to the Forums.
The Desert-rose (Adenium sp.) in the OP, certainly does have an unusual form.
One intriguing possibility is that the weeping stems could be a genetic mutation, just like forms of weeping Willow (Salix), weeping Cherry (Prunus), etc. A genetically stable weeping form of Adenium might be a very valuable plant for breeding.
Another less exciting possibility is that the plant has become etiolated from being grown in low light. The OP mentions that the plant now "gets a lot of sun", but it was only recently acquired. Perhaps it had previously been grown in too much shade.
While it is possible that the plant could have a more serious ailment. It looks rather healthy in the photographs. Healthy but 'weeping'.
The genus Adenium, consists of six to twelve species of pachycaul shrubs, in the Dogbane Family (Apocynaceae), native to eastern Africa, southern Arabia, and Socotra. All species and many hybrids are in cultivation, though Adenium obesum (arguably including A. arabicum) is the most widely grown.