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Go to settings > about phone > Status > signal strength > post the dBm and asu numbers here
That's poor network coverage, that's not a phone issue but a carrier/network issue.
http://en.wikipedia.org/wiki/Mobile_phone_signal
-120 is unacceptable
-100 is poor but usable
-80 is acceptable/decent
-60 is good
-40 is excellent
WiFi should not interfere with your network's signal.
Boot on safe mode and check again
http://talk.sonymobile.com/t5/FAQ/How-to-boot-your-phone-into-safe-mode/m-p/348008#U348008
it could be a third party app acting up