For the four rak‘ahs of Sunnah before Dhuhr and Asr, you should pray them as a single sequence, finishing with just one salam. After reciting the tashahhud in the first sitting, you’ll stand up right away without pausing. The Salat or Darood on the Prophet (may Allah bless him and grant him peace) is only recited during the last sitting.

It has been narrated from the Prophet (may Allah bless him and give him peace) that He used to pray four raka’a after zawal (upon the entering of dhuhr time). He would not make salam except at the end [Tirmidhi].

Abu Ayyub al-Ansari (may Allah have mercy on him) narrates that the Prophet (may Allah bless him and give him peace) used to pray four raka’a before dhuhr prayer, when the sun would decline (i.e. upon the entering of dhuhr time). He would not separate them with a salam. And He (may Allah bless him and give him peace) said, “Verily the doors of the heavens open when the sun declines” [Ibn Majah].