need pics of the roots... long story short, if you are having root problems keep res 100% light proof, use aquashield and pond-zyme as a regular part of your nute mix, and dechlorinate your water with tap water conditioner. Any time you get slimed or algae, change the res and rinse off the roots.
don't count on being able to run DWC with nothing for slime control. Your friend at the hydro store gave you bad advice that could cost you your crop. This is understandable because most hydro methods don't have this problem and most people don't have experience dealing with slime. Yes, some growers can run DWC with straight tap water... even put small amounts of organics in the res... and have no problems. They then think it is the same way for everyone, and give bad advice. I think it has a lot to do with your local environment and what you have in the air. One grower may live in a clean home with no pets while another lives in a trashed out trailer with 4 cats shitting and shedding everywhere. Also if I live in CA, and you live in NY, you have different biology in your local ecosystem. I may never see slime.
The important thing is, if you have a slime issue, the slime is something that will always be there ready to gain foothold and bloom in your res as long as you grow DWC. It is not something you get rid of like fruit flies and they don't come back... you will have to continuously run prevention in the res. Some people do sterile like your hydro buddy, but that is a poor choice because it is only effective at low temps (below 70*), has to be frequently dosed, and is more expensive than bennies. Chlorine in your tap water gasses off after like 24 hours, so obviously that isn't going to work. My tap water has chloramine which hangs around longer, but it is not strong enough to prevent slime blooms (although it does kill my bennies, hence why I use tap water conditioner). Bennies on the other hand actually do better in warmer temps and control slime by out-competing it rather than straight up trying to kill it, and doesn't have to be dosed as often.
If you decide not to use bennies, you will have to run a stronger sterilizer than the chlorine in your tap water. If you can keep the res completely dark and around 65*, you can probably get away with plain tap water, but you take a risk of slime again. Also, make sure you aren't using nutes containing organics.
That said... some seeds just grow slow as fuck.