would it look strange if I was alone? ..... ^ as I do feel shame at travelling alone.^ !?!?!?!?!?!?!?!?!?
Why?
I started eating out alone in my mid teens and travelling by myself, including international journeys, when I was in my late teens and have done it ever since. Even though I am married and mainly travel with DH I still go off on short breaks at times, following interests he does not share. My sister and husband sometimes holiday apart for weeks at a time, because he prefers winter sun and she prefers snow.
You say your funds are limited, depending on your interests, and the date of your birthday, why not a weekend break in a historic town? Most historic towns have hundreds of little B&Bs, many in the town centre and you will see lots of other single people about, restaurants will be busy so that you can melt into the crowd and there will be much for you to do during the day - and theatres, and other entertainments in the evening.
If your funds can stretch beyond that, why not a city break in Europe. You can travel by train or car and there are a lot of agencies running short break trips, with travel and accommodation booked and transport between hotel and airport/railway station. Brussels, Ghent, Brugges, Paris, Amsterdam are all just a short journey away.