I only addressed one of your questions.
Yes, new phones from T-Mobile would likely solve your issue, provided it's not due to terrain, a tower being down, or something inside your home affecting your reception. Hypothetically, if something in my other post doesn't apply, you *should* be able to receive T-Mo 4G/LTE service with the Verizon branded 5S (Apple model number A1533, the CDMA/GSM version of the 5S).
Also find out of your account/line is provisioned for data correctly from T-Mo.
We switched from Verizon to T-Mobile last summer, and the iPhone5s we bought when at Verizon have had reception/signal problems since the switch. We were told by T-Mobile that this is because our phones were optimized for Verizon, and so there would always be signal performance issues.
Is this true? Would this be resolved if we bought new phones through T-Mobile?