Dear Le Scaramouche: though I don't know your particular circumstances, I'm going to take a chance here and suggest the following:

1) Finding a place to sing when you have trouble carrying a melody is very difficult. But there are some places. Community choirs are one site. They often are large enough, strong enough, and inclusive enough to welcome singers like yourself who can usually follow a song if supported by strong singers around you. It's basically the theory of strength in numbers. And while most community choirs have as their goal some performances, most also are able to provide the opportunity for singers with pitch challenges to participate fully.   

I often find that once a group sets its sights on performance, then many group members get all nervous and demanding about "doing it right." Some can get right nasty about it. So, make sure the group you choose is willing to include your 'less-than-perfect' singing voice.

2) recreational singing with friends is another option. Though it may seem awkward at first if you and your friends/family haven't made much music together before, it IS possible to gather and have a terrific time singing as a group just for the hell of it. Anything you can think of to reduce the inhibitions that folks might have (what Frankie Armstrong in the UK calls "disinhibitors" and what I think of as an adequate supply of wine) - go for it. It won't be long before you develop your own repertoire of favourites and you'll be looking forward to those evenings of song without the earlier "disinhibitors".

3) Start with a small and easy repertoire of songs that you REALLY like to sing. Choose material that is very accessible to sing and that everyone knows "by heart" (interesting term, that).    Old songs like "Michael rowed the boat ashore" exist for generations BECAUSE they are great songs and easy to learn and sing.   Avoid complicated songs in the beginning. Build.

4) Another suggestion - sing outdoors alone where you can really let rip and just enjoy the sheer sensual pleasure of your voice pouring out with all you've got. It really is a treat and is just a bumped up version of shower singing.

5) Do what my Mom always did. Sing when you're working with loud power tools, or alone doing repetitive (rhythmic) work.   (My mom was a gorgeous singer, but was silenced in adulthood. To get that story, you'll have to read my dissertation!) Her power tool of choice - okay, maybe not choice - was the vacuum cleaner.

6) You may think you "suck at" singing now, but that's irrelevant. Sing for your own reasons. If you're not sure why you sing, find out why, or find better reasons. It matters - for no other reason that it matters to you. Go for it.

cheers, Moon Joyce