I usually have no problems making friends but it seems that I always seem to have trouble keeping them, not because of misunderstandings or anything but because it just gets super awkward for the two of us to be alone together after a while.
I might suggest that the 'awkwardness' you feel in a relationship with other people is related to your feelings about the relationship itself.
After a while a person might change his expectations about the nature of the relationship, perhaps expect more openness, discussions about personal topics that previously where avoided, more trust, more vulnerability, in other words more intimacy.
When one of the partners has such an expectation while the other person does share such expectations, the expecting person might be disappointed, which in turn might lead to the tension and 'awkwardness' you describe.
So I invite you to observe whether your expectations of your relationships change over time, and also does your relationship partners change their expectations over time and perhaps bringing new expectations to an existing relationship.
It's a challenge to sustain friendships, especially as we get older. One thing that I've noticed is that awkward moments occur when there's nothing going on. I've also found that relationships tend to improve when we do things with one another.
One solution to this is actually not that complicated: do something when you're together. You talk about being alone together - what are you doing in that time? If it's just sitting around, that has the potential to be very awkward very quickly, especially for guys.
Go bowling. Go to the nature center. Go to the range. Invite your friend to attend a lecture with you. Attend a worship service together and discuss that afterward. Learn to golf. Help your friend with a project at their house. Do something charitable together. The main point is: build your friendship on shared experiences together.
Concur with @baldPrussian, above...
A few years ago I found myself going to meditation meetings during which about 20 minutes would be spent "in meditation". The background here isn't important, except to say that this particular meeting only lasted about a year and I hadn't "meditated" before or since.
The biggest benefit that came of this practice, for me, was that I learned to quiet my own thoughts, not silence them outright, but cut the roar in my own head, to a din -- I learned to be with myself, which in turn allowed me to be comfortable in "awkward" moments.
Just another tool...I hope this helps...