The usual treatment for recurrent folliculitis after an initial course of doxycycline would be the use of an antibiotic in a different class. There are several to choose from, but I would usually chose an antibiotic that is better against Staph, such as cephalexin/Keflex or Augmentin. If there is purulent drainage, then a culture may be obtained, but it is also common that folliculitis may not have pus or fluid that can be readily cultured, so treatment is oriented towards what is commonly seen. In looking at the pictures, I do not see any pus or fluid that would generate an accurate culture.
The swelling of the lymph nodes, by itself, would not alter the treatment, nor the rapidity of the swelling. It is normal for lymph nodes to swell in response to infection, and they can swell very quickly. The concern would be if the infection settles into a lymph node and is causing a focused infection in the lymph node with a collection of pus in the lymph node. If pus is collecting in a lymph node, it may need to be drained in addition to the antibiotics. This would be unlikely in your cases for a couple reasons. First, it usually takes longer for the infection to settle in the lymph nodes and generate any significant amount of pus. Second, it would be very unlikely that three different lymph nodes were similarly affected at the same time. Since you have had three lymph nodes swell quickly, it is far more likely that this is reactive swelling of the lymph node.
So, in this situation, it would be appropriate to get the prescription for antibiotics that was called to the pharmacy. There is no need to be seen again in an Urgent Care Center.
If I can provide any additional information, please let me know.