If you dont have access to a guitar or piano, Get yourself an electronic keybord. Even if it is a childs toy it will still be good enough to train your voice.
Does the tuner that you are using generate the pitch you are trying to match? It is better to hear the pitch that you are trying to match.
Most tuners are very accurate. No matter who you are and how good your voice is the tuner will wobble a little.
Once you get the pitch even if you have to slide to get it. sing the same note over and over on that pitch until you can start the note withou sliding to it. At first use a note that is a little higher sounding that your speaking voice.