Do I need a full-frame camera for low-light photography?
No. It's not essential, but usually helps, all else being equal.
The ongoing improvement in sensors means that the latest APSC sensors are about as good in low light situations as a full-frame sensor from a few years ago.
The DxO Sports/low light ISO ratings listed below purport to allow comparison between various sensors. Some care is needed. I own a Nikon D700 and a Sony A77. As a picture acquiring system the A77 utterly trounces the D700. As a low light photo taker the Nikon, with a notional 1.5 stops better performance, in practice is utterly superior to the A77. So, te results below may be what the measurements show - but the differences may well appear larger in practice.
As a useful comparison aid, DXO's "Sports" / Low light ISO sensor figures lists sensor achievable ISO sensitivity for a given set of test conditions for a wide range of cameras.
The APSC Nikon D5300 scores 1338 ISO, whereas the full frame Sony A900 scores 1431 ISO and a Canon EOS 5D scores 1368 ISO. The 2009 Phase One P40 with a larger than full frame sensor scored 1307 ISO. The 5300 is arguably the best low light performing APSC camera ever sold - but it's superiority relative to your NEX-5R is minimal. The 5R has a DXO low-light ISO rating of 910 ISO. In stops this is log_base_2(1338/910) = 0.56 stops.
The best ever performing full frame camera is the aging 12 MP Nikon D3s with a DXO rating of 3253 ISO (!). Compared to your 5R that's log_base_2(3253/910)~= 1.8 stops.
At $US2500 on up used on ebay, that's a lot to pay for the gain, even though it would be nice to have.
A 12 MP Nikon D700 / DxO ISO 2303 / 1.3 stops, can be had for under $US1500.
Note that these results are normalised to 12 MP. Scaling factor is sqrt(Sensor_MP/12) so if eg you have a 24 MP sensor then the actual result will be sqrt (24/12) - sqrt 2 = 1.1 x worse than shown if the full 24 MP image is used. eg if the DXO sensor high ISO figure was 2800 ISO that is for the image scaled to 12 MP. The ISO value that gives the same noise result when the 24 MP image is used ~= 2800/1.4 = 2000 ISO.
Do NOT look at the top right of the table or your life will be ruined :-).
Only owning a Sony A7R will restore it.
The difference between your NEX and the A7R is about 3:1 or about 1.5 stops.
The difference is useful, but the large $ difference would make such a radical change very hard to justify.