You can install separate clips & straps onto your existing pedals. Try leather straps; they're far sturdier than nylon/fabric straps. Another step up would be dual-gate metal clips, which are wider and cover a greater area of your foot. Another step up would be double straps (or two sets of single straps), then finally quad-gate clips.
As for easy adjustment, I've never known how to adjust my straps while riding a fixed gear bike (I need two hands and obviously can't do it while continuously pedalling), so instead I got a strong double strap & quad-gate clip setup (MKS Alpha-Fit double straps and All-City Fourgate Plastic Clips and I use a set-and-forget method. Works for me.