Hello & welcome,
Improper placement or making of the bridge, creates the problem which you are facing & that is food collection.
Food accumulation around tooth or under the bridge causes decaying or caries in the abutment tooth (support tooth). And if you are getting the feeling of tooth going slightly down on application of pressure, then that particular tooth has been decayed to pulp level & there is abscess
formation around the tooth.
Abscess present around the tooth gives cushioning effect on application of pressure along with pain & swelling.
If this is the case, it can definitely be treated with Root canal treatment
followed by placement of new bridge with proper crown preparation by dentist
Definitely the role of antibiotics & painkillers is vital here as there is infection along with abscess formation.
Yes you should go for making new bridge because doing root canal treatment only will not clear the main problem associated with the bridge ( gap underneath ).
This can only be treated with repeated crown cutting & impression taken with due care by the dentist.
If there is food accumulation around wisdom tooth which thereby infect it, you should go for x-ray & even extraction of the same tooth if dentist feels that prognosis is poor.