Hi Max and welcome to Chordie,
Further to KAP's comments you can also use a capo to change the guitar's key. Some songs and styles of playing just do not sound right using different chord shapes. I perform one song with the capo on the 7th fret just so I can keep the open Am key chord shapes for the song to sound right, although I need it in Em to suit my voice.
As for your singing. Like any other disipline it takes practice, practice and even more practice. It is a good idea to record yourself singing and play it back so you can then listen critically and look for improvement. Also your recorded voice is the one everyone hears except yourself. You hear yourself internally and not so much through your ears so it is usually very different to what you expect.
Roger