First, the principle of effective writing is to shorten your sentences whenever possible - especially when you have problems with writing proper English sentences.
Phones would be constantly ringing, in the midst of persistent deluge of e-mails.
Since we were located above a branch, people would often come to us in person whenever they had a problem.
I had to change the mood to subjunctive from the past tense:
Phones were constantly ringing ...
The past tense is for describing a particular point in time in the past:
Yesterday, the phones were constantly ringing.
You could still use the plain past tense to describe the whole block of the, say, two years while you were employed there. However, it is more precise to have a phrase that could be freely glided around any point in time during that block of two years. Rather than the whole block of the two years, including the hours when everyone would be asleep.
To simulate a situation that is infinite or unbound to a particular time, in the past, you would abstract it into a subjunctive modular phrase:
Phones would be constantly ringing.
During my employment at the branch, phones would be constantly ringing.
Especially when you had already abstracted this phrase into its subjunctive unbound module:
people would often come to us in person