@Susurrus, your description of empathic gifts , which is amplified clairsentience, is fairly similar to mine, but I also have clairaudience so sometimes I’ll hear things that are “not there” along with the feelings. I hear it in my head just as if I was talking to myself in my head, however the voice is slightly different. I can’t really tell you how, I just know it is not my inner thoughts. And if I ignore these messages enough, the spirit/deity trying to get my attention will make my ears ring until I listen (ask me how many times I’ve had to learn this lesson).
This will most definitely block you! One of the first things I did and I encourage other “newbies” is to learn grounding and protection spells. These are vital. The key is you want to protect yourself from any intentional, or unintentional, negative energy directed towards you, whether it is from a negative spirit or a clumsy person that said something that inadvertently sent negative energy your way.
There are many ways you can do this and they vary based on what you are doing and what you need. As an empath, I practice veiling daily. The head is the center of the spiritual self and divine communication. Hair is the spiritual “nervous system” of the head and receives/gives energy all day, both positive and negative, which can be very draining. By veiling, or covering the hair, it is a way to calm or regulate this “nervous system” and how much energy it is exposed to. This is also why some cultures such as Native Americans don’t cut their hair. You can use anything as a veil - I usually use bandanas folded in half but have used scarves, have a few lace veils for more formal occasions, and a special black lace mantilla veil I wear during rituals. Each morning I cleanse them, shield them (using the spray below), and ask them to protect me as I tie it around my hair. At night, I take it off, shake it out three times away from my altar, cleanse it again, fold it neatly while thanking it for its protection and place it with it’s buddies in my veil basket.
@MeganB wrote a really good post about how to figure out if you are an empath: Are you an empath? ✨. She discusses common signs and how to shield and protect yourself. @SilverBear gave us this Empath Shielding Spray.
As for connecting with a deity, most of them thrive on attention (some more than others lol). The only time you will offend them is if you are disrespectful. If you approach them with dignity and respect, they will be fine. They understand we are human and will not get mad at us for being said humans. Communication is key. If a deity is pleased, they will tell you. If a deity is displeased, they will tell you so you can reflect on it and make amends. It is also vital to vet your deities to ensure they are not imposter spirits but I won’t get into those details here.
I will say there is a difference between worshipping a deity and working with one. Anyone can worship any deity, you don’t need their permission. You can set up an altar dedicated to them, leave them offerings, ask them for help, and say prayers to them. When you switch to working directly with a deity, you have to approach them and ask if they want to work with you. Some may say no, and it could be for a myriad of reasons. Also note that if you believe a deity is reaching out to you and you are either not ready or do not want to work with them, you can say no, just be respectful. Working with a deity is a lot of work. You not only have the worship part, but your relationship goes beyond that into a mentor/mentee state (or at least this is my experience). They will be there to push you into uncomfortable situations to make you grow (Hades loves his shadow work 
). There will be lots of research, learning new tools and skills, and sharpening your clairsenses. Sometimes as your relationship develops, you realize that you are not compatible with said deity. Again, perfectly ok and normal. Mutual communication is key in the relationship with deities. And you still shield and ward before making contact with your deity so you don’t bring in unwanted visitors.