Wow, some people on this thread are being a little tough...
OP: I have very little experience in this, but it sounds like you match the qualification requirements for the jobs you really want. You also have experience in the field. Yes, it may have been voluntary or while you were not qualified, but you will still have learnt a huge range of essential skills. Perhaps you need to work on your interview skills a bit more, as well as fine tune your CV etc.? Your previous university may be able to help with these - many universities offer careers support to recent graduates.
Make sure you are tailoring each application to the specific job - so go through their specification and make sure you have clearly demonstrated the skills and experience they require. E.g. if they say they need someone with experience working in a caring environment, say 'I gained experience of a caring environment while... '. Sometimes, you really need to make it obvious that you match 100% their criteria in your CV and covering letter.
When it comes to interviews, again make sure you know their requirements off by heart, and have a couple of examples for where you demonstrated each. Sound genuinely interested in the job, and if relevant remaining within the company/ department for a significant period of time (e.g. do some research beforehand on potential opportunities for promotion or gaining additional responsibilities - then ask for more details on these at interview, although make sure you don't sound like you're trying to take the interviewers' jobs!).