Samsung Galaxy S3 SCH-I535, rooted, won't switch over to T-Mobile
SCH-I535 is not a GSM phone so will not work with any 2G or 3G T-Mobile tower. Also that phone only supports LTE frequencies of 700 MHz &amp; 1900 MHz, neither of which are bands that T-Mobile use to any significant degree. Instead T-Mobile uses 1700 &amp; 2100 MHz. 

T-Mobile is however in the process of rolling out 700 MHz &amp; 1900 MHz, so you might get data in the near future however you will not generally be able to make calls as that is currently handled by the 3G networks. And even where VoLTE is available this phone will may not have the software update to support it.

Update: Despite not being listed in the specs of Samsung's website, nor GSM Area, or Wikipedia SCH-I535 does infact support GSM beyond LTE. It appears that Verizon pushed out an update with 4.1.2 Jelly Bean which enabled support for GPRS/EDGE/GSM 850/900/1800/1900 MHz and HSPA/UMTS 2100 MHz (band 1). T-Mobiles 3G support requires AWS-1700/2100 (band 4), however they do support 1900 MHz for GSM/EDGE 2G service.

So you can get EDGE but not HSPA, although you may soon get LTE depending on T-Mobiles rollout.