Hi Iry

If you're remembering your dreams it means you're waking up in the middle of a REM cycle which is why you're also waking up tired. The body functions much better if it wakes up at the very end of a REM cycle because by that stage it's already preparing to wake up, but if you're waking up in the middle of a cycle you're body is still in sleep mode and you're going to wake up groggy and be tired and sometimes lethargic all day.
I used to have the same problem but it's not actually that hard to fix. Firstly realise that with sleep, quantity does not necessarily equal quality - our bodies are lazy and if we sleep 10 hours each night our bodies just think "well I've got 10 hours to sleep so I'm going to give you lousy sleep." Whereas if you sleep only for 6 or so hours a night the body has to work harder to do all it's maintenance work while you sleep and you end up getting much better quality sleep.
Secondly it's important to time when you wake up if you don't want to feel groggy. This can take a bit of experimentation but isn't hard. If you currently get 8 hours sleep a night try setting your alarm clock half-hour earlier so you get 7.5 hours sleep, hopefully this will wake you up
before your next REM cycle starts and your body will be more awake than asleep. You'll know when you've got the timing right because you'll wake up feeling alert, not tired, and a lot of the time you won't remember your dreams anywhere near as vividly, if at all.
Another trick is to keep the time you get up each morning the same, but vary the time you go to bed depending on how tired you are. I do this and it works really well. Since I get up at 6:30am each day I sometimes go to sleep at 11pm if I'm tired, and if not I'll stay up until midnight or later. My body knows it has to be up and about at 6:30 every day so it tells me when it wants to rest.
I'd recommend trying setting your alarm clock back half hour and getting up at the same time each morning and seeing how that works for you. If the half hour makes you more tired than before try intervals of 15 minutes forwards and backwards until you find a time that works for you.
Also if you're sleeping badly once you do get to sleep, and you're not already doing these things, try to incorporate regular exercise and sunlight into your day as these things increase your circadium rhythm which means you'll be able to stay awake longer and you'll sleep more deeply. Having a warm shower (not hot - hot speeds up the circadium rhythm making you more alert) before bed with the lights off also works well if you have trouble dropping off. And remember to keep lights to as minimum as possible just before bed and during sleep as light will prevent melatonin production, making it hard for you to get to sleep.
I hope that helps