We moved into a house that had a garden like that. We started to take up the turf ourselves by cutting it in rows and then liftin it with a spade but it was a pain. We have about a 20ft by 10ft lawn and we asked a local company for a quote to get all the rest of the grass up and returf and they quoted �260 which i thought was a bargin. They did it last September and i cut it at the weekend and i have the most wonderful flat lawn!!!
We tried to get rid of the weeds with weed killer but they always come back. The gardeners went through the soil and picked out the weeds. It really is the only way
We put up with the lawn for about 18 months while we got the house sorted. I know your on a budget but i thought that �260 was better than me having to do the work!!
Obviously it may be a different price for yourself ( we live in West Yorkshire). In the meantime i would just keep it short so it looks better
Hope this helps.