Lion,
The Canon 400mm f/5.6 is a popular budget lens. It is maybe £650 on UK ebay (about £1250 new). It does not have image stabilization, but as long as you keep the shutter speed up around 1/640 or above, you should be fine with practice. Or use a monopod. I understand it takes a 1.4x teleconverter well too. On you 350d, 400mm would be 640mm equivalent field of view, which is a respectable focal length for birding. it is also light weight for a long lens at 1.2kg (without tripod collar attached).
Going from 300mm to 400mm will give you a 1.33x boost. You could take an existing, uncropped, photo you have and remove 25% from the height and width to see what that looks like. For example, a 4"x5" photo would become a 3"x 3.75" field of view.
The next step would be the Sigma 150-600 Contemporary (sometimes just called the "C") or the Tamron 150-600 G2 (the G2 is much better than the original version of the lens). They are both f/6.3 at the long side, but work very well in good light. If it is often overcast or gloomy where you shoot, then an f/6.3 lens might be a bit more challenging. I see on amazon uk, the Sigma is £700 new and the Tamron £1050 new. These are heavier lenses, closer to 2 kg. They will give you a 960mm field of view on your camera.
Going from 300mm to 600mm is a 2x boost. Take an existing uncropped photo, and remove half the width and height to see what it would look like.
Are those prices anywhere near where you were looking? There are older models of 150-500mm or similar from Sigma, but they are very heavy and the image quality is not up to the lenses I just mentioned.
I am a Nikon shooter, so I have not used any of these on Canon. I used to use the Tamron G2 on Nikon and was happy with it. I know several other bird photographers who use the Tamron 150-600G2 on Canon and like it quite a bit. Basically, the choice comes down to : 150-600 for reach, or 400 f/5.6 for lighter weight and faster speed.
The 350d is 8 MP, so you do not have a lot of room to crop down, so the 150-600 might be a better choice. I think the 350d likely tops out at 1600 ISO, so the faster 400mm f/5.6L might be better to keep the ISO lower. I'm not sure how the 350D does with higher ISOs. Sorry to waffle around like that, but I am not sure which way to go for the 350d. On the 7d, I know the 150-600 works fine at moderate ISOs to make up for the higher f-stop.
For these heavier lenses, I recommend using a sling strap, such as the Black Rapids Sport sling strap. The key thing is it slings around one shoulder to keep the weight off your neck. It also has an under-arm strap to keep it from creeping up onto the side of your neck. They work very well, but are a bit pricy. There are many knockoffs, but most do not have the underarm strap. A monopod can also help, especially if you want to keep the lens focused on a target for some time, such as waiting for a bird to take off from a perch.
Price disclaimer: I am searching from the US, so those prices might be off a little, as sometimes I only see shippers who ship to the US. The ebay price is from a single quick search and I cannot represent the quality of the specific product I saw.
Marc