You can never get rid of all genomic DNA contamination. One way to measure it if you are anal retentive about it is to have a pair of primers that both lie in the same exon - do a real time experiment with/without RT in the cDNA synthesis step - theoretically you shuold get no signal from the no-RT control. The delta Ct between the no-RT and +RT will tell you the % genomic DNA (of course, there is variation in the RT efficiency), but you'll still get the idea.
You cant store undisrupted tissue in Trizol, the RNA will degrade. Either homogenize it in trizol and store it or snap freeze it or store it in RNAlater - dropping a piece of tissue in trizol and NOT immediately disrupting it is a very bad thing - slow lysis and release of nucleases. That is absolutely your problem. If you are having lengthy dissections and accumulating sample, RNAlater is probably the safest way.
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