The so called "R" interface on Terminal Adaptors are unregulated and as such the quality varies widely.
Firs try plugging a phone directly into it. If this has the same problem, then the LInsys is simply defective and should be replaced.
First try another Linksys, there may simply be a problem with this unit. If another LInksys of the same model has the problem when connected to your wiring, it is most likely that the POTS jack on the Linsys does not use a transformer to keep the cost down. My guess is that you have LOTS of phone wiring. To keep noise (hum) down in this situation requires three things
a) Good balance in the analog "line" (e.g. in the Linksys TA) usually by using a transformer design
b) Good balance in the installation. This means that each connection in the system must be equivalent on each side of the line. If there is corrosion on the connection of one side, but not the other, this creates an imbalance and that cause noise pickup.
c) Twisted pair cabling.
If replacing the Linksys with something else cannot be done (at least as a test) then here are some things to check that MAY help
1) Check the connections on all the telephone wiring (eg at all junciton blocsk and at each jack. If there is corrosion or anything does not look perfect re-do the connection.
2) Disconnect wiring going to unusused locations.
3) If you have any phone (such as older trimline and princess phones) that send power over the second pair (yellow + black or white/orange and orange/white) disconnect this (these) power supply.
4) If your facility is wired with the old obsolete Red/Green/yellow/black "quad" inside wire, you should replace it. You may not need to replace it all - start with the longest runs first.
You want to use 4 pair solid twisted pair (you may only need 1 or 2 pairs now, but it makes no sense to install 2 pair cable). This will have 2 pairs of wires, each pair twisted. The pairs are:
Pair one
White with Blue stripe (Tip)
Blue with White strip (Ring)
Pair two
White with Orange stripe (Tip)
Orange with White strip (Ring)
Pair three
White with Green stripe (Tip)
Green with White strip (Ring)
Pair four
White with Brown stripe (Tip)
Brown with White strip (Ring)
Use a minimum of Cat. 3 cable (solid twisted pair), though it makes to just go ahead and install Cat. 5e
Good luck