By low signal are you just looking at the bars or are you actually pulling in the dB of your signal. The bars are just a rough representation of the actual signal coming to the phone. You want to check the signal strength dB Settings>About Phone>Status Information>SIM card status. You should see where it says signal strength. What is that number when you are in your apartment. If you are in the area of -110dBm then you are going to start experiencing dropped calls on the cell network.
Also Verizon should offer some sort of signal booster for customers, I would call them and inquire about that. Make sure you call customer retention and be like hey I am going to have take my business else where unless someone can get this fixed. Thats what I do with sprint/tmobile and just about any other utility company. You get some killer offers, just remember to be nice and courteous to the person. They will usually go out of their way to help a concerned and frustrated customer.