Here's the dilemma for someone in your situation: Upgrading the camera will have much less of an impact than it could when still using your current lens, yet upgrading the lens when still using your 350D will limit the improvement as well. I think the lens needs the improvement first, and here is why:
- Lenses with more than a 3X ratio between the shortest and longest focal lengths have many design compromises. In addition to distortion and chromatic aberration, narrow apertures are common with these lenses. For not much more than the EF 24-105mm f/4L IS you are looking at you can buy the Tamron AF 24-70mm f/2.8 Di VC. It is a little sharper than the original Canon 24-70mm f/2.8 L and is both sharper and a stop faster than the 24-105. Your image quality will improve considerably, even with the 350D.
- Although there has been a lot of improvement in high ISO/low noise performance in low light, in the Canon realm most of it at the sensor level has been on the full frame side of things. I own a Rebel Xti/400D, a 50D, and a 7D. In terms of low light performance, there isn't a lot of difference between any of those APS-C models when compared to the clearly superior 5D Mark II. This is particularly true when I use the NR tools in today's software versions on RAW files captured with the older bodies instead of comparing the output from the newer models to shots from the older cameras processed by the older versions of the software. Canon traded the improvements in their Digic processors and firmware/software's noise reduction ability for the smaller pixels used for increased resolution. Your 350D has a sensor with pixels that are 6.4µm wide, comparable to the 6.4µm pixels on the 5D II. The current crop of APS-C Canon bodies, from the Rebel T2i to T5i, the 60D, and the 7D are all based on the same sensor with 4.3µm pixels. The biggest difference between those models are the focus systems, handling speed, and video capabilities - not the basic image quality. If you are intent on upgrading to a Canon APS-C body, I would encourage you to wait until Canon releases a new body with the next generation APS-C sensor. It is long overdue!¹
To get really good low light performance in an environment like your example picture, you're going to have to forget zoom lenses altogether and go with a fast prime lens and a full frame body. Other than the sweet spot around 50mm (due to the way lenses can be designed for cameras with the typical flange distances of DSLRs), wider aperture lenses get expensive very fast. The further away from 50mm you get, the more expensive lenses with comparable apertures become.
If you are serious about improving your low light performance capability on the budget of what an EF 24-105mm f/4 L IS costs, I would suggest looking at used FF bodies like the original 5D and fast primes such as the EF 85mm f/1.8 or the EF 100mm f/2. For 50mm I like the f/1.4² and feel the faster auto-focus and more usable manual focus as well as the durability are worth the difference. Others feel the f/1.8 is a better value. Be sure to buy from reputable dealers like B&H or through places like amazon marketplace where you have recourse if someone sells you a lemon.
¹ Canon's release of the 80D in 2016 finally gave Canon shooters an APS-C sensor with better low light performance.
² When Canon introduced EF 50mm f/1.8 STM they corrected most of the shortcomings of the older EF 50mm f/1.8 II. I would even recommend it over the EF 50mm f/1.4 for the vast majority of users (including myself if I were in the market for a new 50mm prime for less than $1,000).