Hello Everyone!
I've just started running this week and am trying to build up slowly so I've been walking a good bit too. Today I have shin splits in my right leg only!
Any advice... I don't want to lose my momentum... but I don't want to overdo it and end up in too much pain.

Can any of you more experienced runners help?

What shoes are you running in and where did you buy them from? There are some exercises you can do to strengthen your shins such as walking on your heels for 5 mins and make sure you stretch out your calves both the soleus and the gastronemus but you need to identify the underlying cause. Running on surfaces other than roads and pavements can also some relief.

dollymix6... No experience at all with shin splints but welcome
Check some older threads which focus on that.