Good effort, sucks you had those tire issues but the experience counts and the next ride should be better.

I wouldn't think there's a problem with a 28c tire in that rim, however, there might be a problem with tire clearance in your frame.

When dipping the tube in water, make sure you dip the valve as well, sometimes the tube itself would have no punctures and air is lost due to a defective valve.

When replacing a tube, after mounting the tire, it's a good idea to inflate the tube just a little, and then double check that the tire bead is fully seated on the rim bed, before inflating the tube to desired final pressure.

Someone else might chip in with tire recommendations as i have not used 28c tires.

Good luck