The reason you are in roaming is exactly why you stated, you broke the antenna. Without the connection there, you will never pick up a very good signal. Typically you can find the backplates for phones on Ebay pretty cheap, so I would start there.
Do you mean the two little gold pins next to the SIM card?
I don't know what those are, but they do not interact with the back in any way and a new back will not help. I think that they are test points. If you have broken one off, you must have been trying to do something other than removing the back and battery. At this point you have several options; all will cost you money.
- Do nothing.
- Call Samsung. get a cost estimate, and then send your phone to Samsung for repair.
- Call Verizon, explain your problem, and ask if they will sell you a reconditioned GS3 or any other phone at a reduced price.
- Try to find a local repair shop that will fix it for you. Many local shops are more expensive than Samsung repair.
- Buy another phone on Craigslist, Ebay, etc. Buyer Beware!
- Sell your phone for parts to recover a little of your costs.
After looking online at various Galaxy S3's with the back removed (mostly international version), it appears that only the Verizon version has those gold pins.