Why do you want to be a vampire? Being a vampire in the vanilla game is terrible, you will be refused services. Vampirism is only worth it with mods. So if you have a pc, then I suggest you look up: Vampire Embrace and Vampire Hunger. They are both compatible and add much to the experience.
To answer you question: Try open and closing a dead vampire's inventory screen, sometimes you can get infected.
Sometimes it can take many attempts to get infected.
It's not as easy to accidently get infected as later games.
Searching their bodies does help.
Because each Clan has a questline. Because it's something different/new. Because some of your skills get powerful. Because it's a TES game!
If on PC, yes there are many mods that can enhance the experience.
You can also just make yourself a vampire with the console.
I had a vampire that completed the main quest and built the Factor's Estate on Solstheim. It was on the Original Xbox and she was great fun to play. You just have to design the character right and prepare some things in advance to play as a vampire in vanilla Morrowind.
OP: Save before you try to become a vampire. If it fails, reload and try again. Check the bodies multiple times . You will eventually roll the disease. Make sure your spell is 100% weakness and has a fairly long duration (30 secs should be long enough). Spam the spell while the vampires are wacking on you if it is a short duration spell or it runs out.
If you had your Corpus cured in the Main Quest you cannot become a vampire. You said you finished the game so I assume you were cured of Corpus. If you want to be a vampire you will need to start a new character. Look in spoiler for more info.