For good bed adhesion and anti warp, I use a BuiltTek 6.5x6.5 sheet stuck right to the build plate. All PLA prints stick to it really well, and come off super easy without needing a scraper all the time. If your print room gets cold (>20C [>68F]) try a small (really small) space heater to warm the build plate before you start the print. Rule of them here is to let it heat to where you don't want to keep your hand on it too long. This should let the PLA cool a little slower to keep it from peeling off of the plate. Also use 5mm brims if your print can fit it.
In order to deal with the nozzle dragging: look in the setup settings, where you find the calibration setting. you should find something called "Z-Offset". Go into that setting and click the up arrow. Let the axis home, then gently slide the extruder head to the center of the build plate; lastly, slide the build plate to roughly the center. Take a business card or two sheets of A4 paper and slid them under the nozzle. There should be a little drag on the surface of the paper, but not enough to scratch the paper.
A good note is to do this after a cleaning cycle so that you don't have a little spike of plastic sticking out of the nozzle.
ALSO: Do not turn off Auto Leveling. That is a built-in code that accounts for the low and high spots on the build plate. So if you (like I) have a slightly warped build plate, the software will adjust the z-height as the extruder moves around the bed.
