To confirm you have RNA you can add some RNAse to your agarose sample buffer. (warning - if your lab works with RNA, be sure they're okay with you using RNAse in the gel box). The band-like effect could be degradation of the DNA (into nucleosomes or something like that). Plants can be difficult to work with - are you using young, tender (easy to break open) leaves or tough, old, difficult to work with leaves? For DNA, use the youngest, most tender part of the plant (for RNA you don't have that luxury). And, yes, give us more information about the kit or reagents you're using. Most have RNAse in them. If you have a spectrophotometer, what is the 260/280 ratio? That will tell you protein contamination. And subtracting the A320 will tell you if you have particulates (or phenol, if you used it) in the prep.
DO NOT use a spectrophotometer for quantifying DNA - it will give wildly inaccurate values. Unless performed with kits that use columns to bind the DNA, Plant DNA preparations invariably contain high levels of polysaccharide which confound quantification. Use FLUOROMETRY instead.
DNA is not fragmented - the horizontal band just below the wells is your genomic DNA and it is "sharp," with very little "tailing" indicating minimal degradation.
DNA quantity is low, however, its quality is good. I sure the smear on gel is related to RNA contamination of extraction not DNA fragmentation which can be eliminate using RNase enzyme.