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What you described sounds like a SIM card issue not a phone issue, get a brand new SIM card and report back
If you want to try something else, you could repair the phone with PCC and check again, but the problem seems a faulty SIM card.
Bridge (for Mac)
Alternatives on How to backup Xperias
http://talk.sonymobile.com/thread/36355
Hi renneyleslie, welcome to the community!
Some steps on running the actual repair on either PCC or BFM:
Running the software repair on PC Companion:
1. Install and open PC Companion then select Support Zone.
2. Phone Software Update.
3. Start.
4. Repair Phone
5. Continue
6. Accept data removal
7. Next
8. Wait for prepare
9. Select Phone
10. Follow the connection steps
Running the software repair on Bridge for Mac:
1. Launch Bridge for MAC without the handset connected.
2. On the Main Window select "Xperia > Repair Xperia
Device".
3. Ensure the phone is charged and select "Ok" in the
following window.
4. Select Continue on the following page.
5. Wait until the initialising step is finished the
continue as below.
6 . Select your model of phone from the list that appears
7. Follow the connection steps that appear, ensuring that
you are holding the relevant button down before during and
after the connection phase until the update/repair begins.
This is very likely to smoothen out the OS for you, let us know how it goes.