‘How Great Is Our God’, ‘Here I Am to Worship’, ‘Open the Eyes of My Heart’ and ‘Lord I Lift Your Name on High’. If you want something older, ‘I Love You Lord’ and ‘As the Deer’ are two-minute songs you can learn in one sitting.
‘Goodness of God’ is forgiving to sing, same with ‘Build My Life’ since the melody stays in a comfortable range.
Pick two or three and stick with them for a few weeks instead of rotating constantly. My most meaningful moments have been with songs I’d sung a hundred times, just because I wasn’t thinking about the words anymore.