Is it 'bad' that I have a really terrible memory and always need to look at tabs to play a song? I seem to have no trouble remembering parts of songs, but to sit down and play a whole song from start to finish without looking / cheating is hard for me. What do you think?
Depends on several things, not least of all the particular song. Some songs I can memorize easily - I've played guitar more than 40 years so I have some practice at it. There are some songs whose chord progressions lend themselves to memorization. Others I need a hint or even the entire tablature to play, even if I work at them.
Generally if I play a song often enough, it finds a memory groove in which I will eventually remember it without prompting. What's annoying is that I can still play songs I learned in the 60s and 70s without hesitation, but that new songs I'm working witrh are often much more of a struggle to commit to the memory bank.
I have a good memory for patterns and shapes. I don't get lost easily, I find my way easily in FPS and RPG games, I do well in recognition tests. I remember the location of books in my collection by the pattern of the shelf. That makes it easy to memorize chord patterns.
Of course, I am of the age when memorization was common in schools. We learned to memorize large poems, or pieces of poems, or soliloquies from Shakespeare. That really helped because I can memorize lyrics today thanks to that training.
It's not "cheating" to refer to song sheets or tabs. Probably most of us have a library of 30-50 songs we like to play; if you don't play them regularly and frequently, you simply can't remember each one. Use what works, use what makes sense. Keep in mind that actors today don't memorize whole scripts or politicians whole speeches: they have prompters to remind them of their next lines.
Here's an idea: go to a copy shop and photocopy your favourite songs at 25%, then copy them 4 on a page, double sided and cut them into quarters. You can copy them onto card stock or laminate the paper for better stiffness. This will give you small, easily carried song sheets you can put in your case, put on a music stand or in stick into one of those uke clips.