[i]"sine looks nice, but still off-centered.."[/i]
You dont say where you took the signal from, or what the waveshape setting was..
If the signal was taken from the audio output, and was at the 'sine' setting, then you have a problem somewhere in the audio path - this is a 'soft' square wave, not a sine! .. If this was taken at the square wave setting, then it looks great - and if you can adjust from this to sine, then it looks like you have fixed the problem.
If signal was taken from the audio output, then the offset is not 'real' - it is due to your "scope".. The output is capacitively coupled (C38) and the waveform therefore MUST cross the mid value which should be the 0V / Gnd reference of your 'scope.. The only exception to this (extremely unlikely) is if there was a big DC offset on IC2:6 AND C38 was 'leaking' this DC to the 'scope input.
You do need a 'scope.. The one Thierry recommends is good value and, as he says, perfectly suitable to this type of application.. Without a 'scope you cannot take accurate measurements of signal levels, offsets etc..
A lot depends on how much you are willing / able to spend on test equipment, and on how much electronics 'work' you intend to do. If you plan to do a lot, and can afford it, I would recommend the Hantek DSO1060 (http://www.hantek.com.cn/english/news_list.asp?unid=14) – this instrument combines everything you need (High resolution frequency counter, High resolution DMM which can display simultaneously with waveform, and simultaneous readout of a large number of waveform parameters, 60MHz Dual Chan ‘scope) in one compact unit.
The DSO1060 can display FFT, but it is not a good spectrum analyzer (8 bits vertical resolution is not good enough to resolve low-level harmonics.. Useful for quick checking for gross distortion, but no good for evaluating ‘Hi-Fi’ circuits). My only quibble re the quality of this instrument was the ‘scope probes – their compensation is poor compared to other probes – but a good pair of probes costs nearly half the price of this instrument, and they are good enough – so I am just being silly!
I paid about £350 for my DSO1060, but have seen them on sale now for about $500 .. You can probably get 3 or 4 UT81B’s for this money – If you bought a UT81B, and a high res Frq counter, you will still be paying a lot less.. but if you intend to go further in electronics, my advice is to get good equipment at the start - The DSO1060 is the only 'scope/multimeter/frequency counter you will need - add an adjustable PSU and signal generator, and you have all the test equipment required for almost anything.
Fred.