Yes, a ph meter first to see where the ph is. Lime can be a detriment if it is not needed.
If you find that your ph is OK, then you need to give it some fertilizer. Get some 16-16-16 if you can find it and give your plants a treat. Careful, don't over do it.
My soil has plenty of potassium and phosphorous, so I just boost the nitrogen with some urea that is 46-0-0. It doesn't take much of that to make a big difference in the growth of the plants.
If you are going to add lime, you should try to spread it uniformly on the whole area. The plant roots go everywhere so the whole area needs to be at the right ph.
Plant roots will go quite deep, so they do indeed go into the soil that your raised beds sit on, and will also likely go out into the soil surrounding the raised beds. Did you put a barrier? I think a barrier is a bad thing. What is your natural soil there like? Test its ph too when you get your meter.